PERT Computerized Placement Test

Transcription

1 PERT Computerized Placement Test REVIEW BOOKLET FOR MATHEMATICS Valencia College Orlando, Florida Prepared by Valencia College Math Department Revised April 0 of 0 // : AM

2 Contents of this PERT Review Booklet: General information.... Study tips.... Part : General Mathematics Section... Test for General Mathematics... 6 Test answers.. 6 Tips on how to work missed questions on test.. 7 Part : Developmental Math I.. 8 Test for Developmental Math I.. 9 Test answers.. Tips on how to work missed questions on test.. Information sheets on a variety of Dev. Math I topics.. Practice questions and answers from Dev. Math I Part : Developmental Math II. Test for Developmental Math II. Test answers 6 Tips on how to work missed questions on test.. 7 Information sheets on a variety of Dev. Math II topics. 69 Practice: Signed no., Equations, Graphing, Polynomials... 7 Practice: Variety of topics with answers... 8 Part : Intermediate Algebra. 88 Test for Intermediate Algebra 89 Test answers... 9 Tips on how to work missed questions on test... 9 Valencia College All Rights Reserved of 0 // : AM

3 PERT Review for Mathematics This PERT Review was created to help students to review the major skills that are assessed on the PERT test in order to achieve the most accurate placement into a course of mathematics at Valencia College. If you have not learned the subject matter covered in this booklet at an earlier time, it is unlikely that you will be able to learn it for the first time through this review. No calculator will be allowed on the PERT test. The PERT test does NOT have a time limit. Valencia College All Rights Reserved of 0 // : AM

4 Math Study Skills Tip Sheet. Read your tetbook. Read your tetbook before the topic is covered in class. Make notes on anything you do not understand, so you can get that cleared up during class. It is crucial to master one concept before going on to the net and to stay current with your reading. Read actively - read with paper and pencil in hand - work out eamples yourself - highlight and take notes.. Set up a regular study time and place. Be aware of your best time of day to study. Study two hours for every hour you spend in class. Because most math classes are cumulative, it is better to study for a shorter amount of time more often, then to wait until the day before the test. Study math first if it is your most difficult subject. Study with others - you will learn different approaches to reaching solutions.. Be actively involved. Attend class regularly. Come to class with homework completed, if possible. Speak up when you have a question. Seek etra help, if necessary. Mat h Cent er s wit h f r ee t ut or s ar e available on all campuses of Valencia College. Valencia College All Rights Reserved of 0 // : AM

5 Part General Mathematics All students entering Valencia College should review this section. Some of this general mathematics material will be taught in the Developmental Math I course. For any math class higher than Developmental Math I the instructor will assume their students know everything in this section. NOTE: The PERT test will not specifically test anything in this section but questions on the test will assume you know your basics in order to answer some of the algebra questions. Valencia College All Rights Reserved of 0 // : AM

6 Test for General Mathematics Choose the one alternative that best completes the statement or answers the question. Evaluate: ) 9 6 a) 6 b) c) 8 d) 7 ) 76 a) 98 b) 66 c) 6 d) 6 ) 9 a) 7 b) c) 9 d) ) 8 8 a) 6 b) c) 6 d) 6 Find the area of the shaded region: ) 7cm 8cm 6cm a) 68 square cm. b) 76 square cm. c) 78 square cm. d) 8 square cm. Find the average: Solve: 6), 6,, a) 6 b) c) 7 d) 6 7) For five mathematics tests your scores were 8, 86, 8, 76, 7. What was your average score? a) 8 b) 78 c) 79 d) 7 Choose a strategy and solve: 8) Your car gets about 0 miles per gallon. You are planning to drive to see your friends who live about 80 miles away. How many gallons of gas will you need to purchase to make the trip to see your friends and to return home? a) About 0 gallons b) About gallons c) About 8 gallons d) About 8 gallons 9) In 006 your car cost $,0. In 008 your car cost $7,76. How much did the price increase? a) $06 b) $06 c) None, it was a decrease d) $06 0) While shopping for CDs, you note that the average price is about $8 per CD including ta. You have $6 in your pocket. About how many CDs can you buy? a) b) c) 7 d) 9 ) In 007 you weighed 07 pounds. In 008 you weighed 98 pounds, and in 009 you weighed 8 pounds. How many pounds did you lose from 007 to 009? a) lbs b) None, you gained weight c) 9 lbs d) lbs Valencia College All Rights Reserved 6 6 of 0 // : AM

7 Solve: ) Write this epression in words: = 6 a) The sum of 9 and 7 is 6 b) The difference between 9 and 7 is 6 c) The product of 9 and 7 is 6 d) The quotient of 9 and 7 is 6 Identify a fraction or mied number that represents the shaded part of the figure: ) a) b) c) 6 d) 6 ) a) 7 8 b) c) 7 d) 7 Which diagram represents the number: ) a) b) c) d) 6) a) b) c) d) 7) a) b) c) d) Indicate whether the number is a proper fraction, an improper fraction, or a mied number: 8) 70 7 a) Improper fraction b) Mied number d) Proper fraction Write the mied number as an improper fraction: 9) a) 7 7 b) 9 c) d) 9 7 Write the improper fraction as a mied number: 0) 7 a) 7 b) 7 7 c) 8 d) 6 Valencia College All Rights Reserved 7 7 of 0 // : AM

8 Find the value of n: ) n 6 80 a) 80 b) c) d) 00 ) n a) b) c) d) Simplify: ) 0 a) b) c) 0 d) ) 60 0 a) b) 7 c) 7 d) 60 0 ) 60 6 a) b) c) d) Between the pair of fractions, insert the appropriate sign: < = > 6) 8 a) < b) > c) = 7) 6 a) = b) > c) < 8) 7 a) < b) > c) = Solve: Write your answer in simplest form: 9) A baseball team has played 9 games so far this season. The team won 7 games. What fraction of its games has the team won? a) 9 7 b) 7 9 c) 6 7 d) 7 6 0) Of a family s $8 weekly income, $86 usually goes toward groceries. What fraction of the family s weekly income is usually spent on groceries? a) 8 86 b) 86 8 c) 8 d) 8 Valencia College All Rights Reserved 8 8 of 0 // : AM

9 ) You have three bolts that are 8 in., 6 in., and in. long. You select the shortest of 8 these to join two plates. Which length is selected? a) 8 inch b) 6 inch c) 8 inch ) A broker has an order to sell 00 shares of XYZ Company stock if the price rises another 7 of a point. The stock went up points today. Does the broker see the 6 stock? Add and simplify: a) Yes, 7 b) No, 7 is greater than, so the stock gained enough to sell. 6 is less than, so the stock didn t gain enough to sell. 6 ) 8 8 a) b) c) d) ) a) 77 b) 9 c) 9 d) 6 9 ) 6 7 a) 9 b) 0 c) d) 7 6) a) b) 7 c) 9 d) 7) 7 7 a) 7 9 b) c) 9 d) 6 9 8) 7 7 a) b) c) d) 9) 7 8 a) 7 0 b) 0 c) 6 d) 6 0 Subtract and simplify: 0) 8 8 a) b) c) 8 d) 6 Valencia College All Rights Reserved 9 9 of 0 // : AM

10 ) 8 a) 7 b) c) d) 0 ) 7 a) 9 b) c) 7 d) 7 ) 7 9 a) b) c) 8 d) 6 ) a) 6 b) 6 c) d) ) a) 7 b) 7 c) 7 d) 7 6) 0 7 a) 9 7 b) 7 c) 7 d) 7 7) 7 a) b) c) d) Solve: Write your answer in simplest form: 8) There were 8 yards of wire on a spool. After a customer bought from the spool, how many yards were left? 8 yards of wire a) 8 yards b) yards c) 8 yards d) 8 yards 9) Brian was training to run a marathon. During the three-day period before the race he decided that he would train for a total of hours. If he trained for hours on the first day and third day? 9 0 hours on the second day, how many hours would he need to train on the a) hours b) hours c) hours d) 6 hours Multiply: 0) 8 a) b) c) d) 8 Valencia College All Rights Reserved 0 0 of 0 // : AM

11 ) 8 a) 88 b) 88 c) 8 d) 88 ) 7 a) b) 8 c) d) ) 9 a) b) c) 6 d) 08 Divide: ) a) 6 b) c) 7 d) 7 ) 8 a) 8 b) 6 c) d) None of these 6) 7 a) b) c) d) Solve the problem: 7) Jim has traveled of his total trip. He has traveled 0 miles so far. How many more 6 miles does he have to travel? a) 6 miles b) 86 miles c) None of these d) 0 miles 8) A bag of chips ounces. A serving size is of an ounce. How many servings are in the bag of chips? a) 8 servings b) 9 servings c) servings d) 6 servings Write the decimal as a fraction or mied number in lowest terms: 9) 0. a) 96 b) c) 7 00 d) ).6 a) 9 b) c) d) 68 Valencia College All Rights Reserved of 0 // : AM

12 Write the number in decimal notation: 6) Eight and seventeen hundredths a) b) 8.07 c) 87 d) 8.7 The following sentence involves decimals. Write the decimal in words: 6) The weight of a full grown Great Pyrenees dog named Simba is.86 pounds. a) One thousand fifty-two and eighty-si tenths b) One hundred fifty and two hundred eighty-si thousandths c) One hundred fifty-two and eighty-si hundredths d) One hundred fifty-two and eighty-si thousandths Write the number in decimal notation: 6) A piece of paper is thirty-two thousandths of an inch thick. a) 0.0 inches b) inches c) 0. inches d) 0.00 inches Identify the place value of the underlined digit: 6) 0.97 a) Thousandths b) Ones c) Hundredths d) Ten-thousandths Between each pair of numbers, insert the appropriate sign: < = > 6) a) > b) = c) < 66) a) = b) > c) < Rearrange the group of numbers from smallest to largest: 67).0,.00, a),.0,.00 b).0,.00, c),.00,.0 d).00,.0, Give an appropriate answer: 68) Last summer, your average daily electric bill was for 9.0 units of electricity. This summer, it was for 9.0 units. During which summer was the electrical usage higher? a) Last summer b) The usage was the same for both summers c) This summer d) Not enough information to determine Round as indicated: 69) 8.68 (nearest hundredth) a) 8.6 b) 8.6 c) ) 8.7 (nearest tenth) a) 8.8 b) 8.7 c) 8.6 Valencia College All Rights Reserved of 0 // : AM

13 7) $ (nearest cent) a) $0.07 b) $0.00 c) $0.08 d) $.00 7).9% (nearest tenth) a) % b).9% c).9% d).8% 7) $,.7 (nearest hundred) a) $,0 b) $,00 c) $,00 d) $,00 Write the ratio in simplest form: 7) to 6 a) b) c) d) Indicate whether the statement is True or False: 7) 0 76). is to 0.6 as.6 is to. Change the percent to a fraction or mied number. Simplify if necessary: 77) 0% a) b) 0 c) d) 78) 00% a) 6 b) 0 c) d 79) 7 % 0 a) 7 0 b) c) 7 d) 0 Solve the problem: 80) What is 0% of 00? a) 0 b). c) 00 d) 8) What is 0.% of 00? a) 6 b) c) 60 d) 600 8) Compute 0% of 0 trees: 8) Compute a) 9,90 trees b) 99 trees c) 99,00 trees d) 00 trees % of 8 feet: a).8 feet b) 0.0 feet c) 8 feet d) 8. feet Valencia College All Rights Reserved of 0 // : AM

14 Find the percent of increase or decrease: 8) Original value: $0 New value: $8 a) % increase b) % decrease c) 0% increase d) 0% decrease 8) Original value: $0 New value: $ a) 70% decrease b) 68% increase c) 80% decrease d) 70% increase Find the mean of the set of numbers: 86) $, $, $6, $, $, $, $6 (Round to nearest dollar) a) $7 b) $ c) $ d) $8 87).7, 6., 7.,.7,. a).8 b).0 c). d).66 Use the table to solve the problem: 88) Moons Average distance from Geo I (km) Diameter (km) Time of Revolution in Earth years Luna Luna Luna 90,000.6 Luna 9, Luna 97, What is the time of revolution around Geo I of the moon Luna? a) 0.77 years b).8 years c).86 years d) 7 years Valencia College All Rights Reserved of 0 // : AM

15 89) Lake Area Long Lake 8 Big Horn Lake 886 Green Lake 80 Pokagon Lake 890 Thomas Lake 8 How many times as large as the smallest lake is the largest lake? a) 0.09 times b) 80 times c). times d) 0 times 90) Use the following table to determine the minimum payment on a credit card bill: Balance $0 - $ $.0 - $0 $0.0 - $000 $000.0 and up Minimum payment that must be paid Full balance $ 0% of balance $00 + % of balance greater than $000 What is the minimum payment if you have a balance of $00? a) $00 b) c) $ d) $0 Change the given quantity to the indicated unit: 9) 00 seconds = minutes a) 9 b) c) d) 9) 80 inches = feet a) 60 b). c) 0 d) Find the perimeter: 9) 8 yards yards a) yards b) 6 yards c) 8 yards d) yards Find the square root: 9) 9 a) b) 9 c). d) 7 9) a) b) 7 c) d) Valencia College All Rights Reserved of 0 // : AM

16 Answer key for Part General Mathematics section: ) D ) D ) D ) A ) A 6) D 7) C 8) C 9) B 0) C ) A ) A ) D ) B ) B 6) C 7) C 8) B 9) D 0) A ) B ) A ) D ) B ) A 6) B 7) C 8) C 9) B 0) B ) B ) A ) C ) C ) A 6) A 7) B 8) B 9) D 0) C ) D ) B ) D ) B ) B 6) C 7) B 8) A 9) B 0) D ) C ) D ) B ) A ) B 6) B 7) D 8) C 9) D 60) C 6) D 6) C 6) A 6) A 6) C 66) B 67) C 68) C 69) A 70) B 7) C 7) C 7) C 7) C 7) FALSE 76) TRUE 77) D 78) D 79) B 80) A 8) A 8) B 8) A 8) C 8) A 86) D 87) D 88) B 89) C 90) D 9) B 9) D 9) D 9) D 9) C Valencia College All Rights Reserved 6 6 of 0 // : AM

17 BASIC WORD PROBLEMS Following these steps will help you get through a word problem:. Draw a picture that matches your information.. Put all information from the problem on your picture. (In English - Not Algebra) If you can tell what the problem says by looking at your picture, then you have done an ecellent job. OPTION: Make a chart for all your information. (Great organizational tool.). To know that you really understand the problem, try putting in a reasonable guess of the answer and then figure out if it is correct or incorrect. You could continue to guess at the answer until you get it correct, but eventually guessing will take up too much of your time.. Using algebra will allow us to find the correct answer without all the guessing. This means that we will use a variable to represent the correct answer in place of the value we were guessing. We will write an equation very similar to the work as when we were guessing. Equation clue words: TOTAL, SUM, or similar words.. Solve the algebraic equation. Check your answer. 6. Answer the original question by writing in sentence format. Eample: At a service station, the underground tank storing regular gas holds 7 gallons less than the tank storing premium gas. If the total storage capacity of the two tanks is 8 gallons, how much does the premium gas tank hold? Premium gas tank Regular gas tank TOTAL (clue word) 8 gallons Holds 7 gallons less than premium tank I am going to guess that the premium tank holds 00 gallons. To check our guess we can calculate that the regular gas tank holds gallons (7 less than the premium tank of 00 gallons). If our guess had been correct the total for the two tanks would be 8. But because 00 and totals to the incorrect value of 9, we need to guess again! Using algebra we can assign a variable (the correct answer) to the amount in the premium tank. The correct amount in the premium tank is P. Therefore the amount in the regular tank (which is 7 gallons less) is P - 7. Now since these answers are correct the total will be 8 gallons. Reasoning: Amount in premium tank + Amount in regular tank = 8 gallons Equation: P + P - 7 = 8 Solve: P -7 = 8 (Combine like terms) P = 900 (Add 7 to both sides of equation) P = 0 (Divide both sides by ) Answer: The premium tank holds 0 gallons of gas. Valencia College All Rights Reserved 7 7 of 0 // : AM

18 WHAT THE WORDS MEAN Addition: Sum the sum of and + Addition: Plus plus y + y Addition: Increased by h increased by 8 h + 8 Addition: Eceeds eceeds m by 6 m + 6 Addition: Added to 7 added to m m + 7 Addition: More than h more than + h Addition: Greater than greater than y y + Subtraction: Difference the difference of k and k Subtraction: Minus minus Subtraction: Decreased by h decreased by 8 h 8 Subtraction: Reduced by reduced by r r Subtraction: Less less m m Subtraction: Less than less than c c Subtraction: Subtracted from 8 subtracted from r r 8 Multiplication: Product the product of and Multiplication: Times times m m Multiplication: Twice twice w w Multiplication: Of half of t / t Division: Per miles per gallon miles/gallon Division: Quotient the quotient of r and 7 r / 7 Division: Divided by divided by y / y Division: Ratio the ratio of z to z / Division: Split into split into n equal parts / n Eponent: Square the square of m m Eponent: Cube the cube of r r Equals: Is The sum of and r is 9 + r = 9 Equals: Result is If you increase m by 7, the result is twice m + 7 = Inequality Is not equal to y is not equal to twice y Inequality Is less than 6 is less than r 6 r Inequality Is greater than m is greater than y m y Inequality r is less than or equal to z r z Inequality p is greater than or equal to w p w Valencia College All Rights Reserved 8 8 of 0 // : AM

19 Multiplying & Dividing Fractions and Mied Numbers Steps:. Rewrite all mied numbers as improper fractions.. If it is a division problem, find the reciprocal of the fraction following the division sign and then multiply.. Multiply the numerators.. Multiply the denominators.. Reduce the fraction, if possible. 6. If the fraction is improper, rewrite as a mied number. Multiplication Eample Division 8 8 Step # 8 8 Step # Step # & # Step # Step #6 Easier Option: Reduce any common factors before multiplying. 8 Step # 8 7 Step # Step #6 7 Valencia College All Rights Reserved 9 9 of 0 // : AM

20 Adding and Subtracting Like Fractions We can ONLY add and subtract the same thing. In the world of fractions the denominators must be the same to have LIKE fractions. Steps:. Add or subtract the numerators (How many units you are adding or subtracting).. Write down the denominator (What you are adding or subtracting).. Reduce your fraction to lowest terms.. If it is improper, write it as a mied number. Eamples with steps # and # only: Eamples that include step # and/or step #: To reduce a fraction we need to find a number that will divide evenly into the numerator and denominator. In the eample above goes into 8 and 0 evenly. Because the / has a value of and multiplying by one does not change the value of the epression, we can drop it from the problem In this eample we have to reduce as in the previous eample. And then write the final answer as a mied number because the numerator is greater than the denominator. Valencia College All Rights Reserved 0 0 of 0 // : AM

21 Adding and Subtracting Unlike Fractions Steps:. Find the common denominator (the number all denominators go into evenly).. Using appropriate identities change all fractions to the common denominator.. Add or subtract the numerators (How many units you are combining).. Write down the denominator (What you are adding or subtracting).. Reduce your fraction to lowest terms. 6. If it is improper, write it as a mied number. LCD is : The smallest number that and both go into evenly. Find the appropriate identities that will give a denominator of. 8 Multiply by the identities to create LIKE denominators. Add numerators, if problem is addition. Subtract numerators, if problem is subtraction. Valencia College All Rights Reserved of 0 // : AM

22 Subtracting Mied Numbers Steps:. Make both fractions into like denominators.. Subtract the numerators. If the value is negative, you must borrow one () from the whole number. The borrowed one () must be changed to an identity fraction with the same denominator as the fraction and then added together. Now you can subtract the numerators and get a positive value.. Subtract the whole numbers.. Reduce the fraction, if possible. Eample 8 8 Rewrite vertically and change into like denominators. 8 8 Since subtraction at this point would create a negative fraction, we need to borrow one from the whole number () to keep our values positive was chosen because we can only add like fractions. 8 Borrow Borrow from and write as an identity fraction 8 0 with the same denominator. Add numerators Subtract the fractions and whole numbers. 8 This fraction cannot be reduced. Valencia College All Rights Reserved of 0 // : AM

23 Working With Decimals Assumption: When we do not see a decimal in a number it is assumed that it is after the last digit of the number. Place value: ones tens hundreds thousands values greater than one. fractional values ten-thousandths thousandths hundredths tenths Rounding off: 6.7 Round off to nearest tenths. The correct answer will be 6. because the digit after the tenths column is or higher Round off to nearest thousandths. The correct answer will be because the digit after the thousandths column is below. Adding / Subtracting: Line up the decimal points to make like columns. When subtracting you should first put zeros in all empty columns. Multiplying: Step : Multiply the numbers as though there are no decimal points. Step : Total up the number of digits after both decimal points. Step : Put the decimal in the answer so that the number of digits after the decimal will be the same as your answer to step #. Dividing: The divisor (number you are dividing by) must be a natural number. Step : If the divisor is not a natural number, then move the decimal all the way to the right. Step : If you move the decimal in the divisor, you must also move the decimal in the dividend (number you are dividing into) the same amount of places. Zeros may be added, if needed. Step : Divide normally. Step : Zeros can be added to the dividend to get appropriate answer. Valencia College All Rights Reserved of 0 // : AM

24 Putting Numbers in Order by Relative Size Comparing fractions: Must have common denominators. Put in proper inequality: 6 Rewrite with common denominator: 9 0 Because 9 is smaller than 0: 9 < 0 Comparing decimals: Must look at common columns. Put in proper inequality:.6.68 Look at each common column individually starting from the left. Tens column: Both are equal (). Ones column: Both are equal (). Tenths column: Both are equal (6). Hundredths column: hundredths is greater than hundredths. Thousandths column: Since we have already found a larger column that determines which is greater, this is unimportant! Therefore:.6 >.68 To compare more than fractions or decimals you can use the same procedure as above to find the largest or smallest value. Valencia College All Rights Reserved of 0 // : AM

25 Percentage Information Per cent age means: Out of 00 or.0 or / 00 Number % of Number = Number % of 80 is what? 0 % of what is? What % of 0 is 0? (.0) (80) = number 0 (.0) (number) = number (.0) (0) = 0 ()(.0)(80) = number (0)(.0)(number) = (number)(.0)(0) = 0 0 = number (.0)(number) = (number)(.0) = 0 number = 70 number = 0 (Divided both sides by.) (Divided both sides by.) Therefore: % of 80 is 0 0% of 70 is 0% of 0 is 0 Percentage is less than one (): 0. % of 7 is what number? / % of 0 is what number? 0.(.0) (7) = number ( / ) ( / 00 ) (0) = number 0.87 = number / 0 = number Percentage is between and 99:.8% of 9 is what number? / % of 0 is what number? (.8) (.0) (9) = number / % of 0 is what number? 7.86 = number ( / ) ( / 00 ) (0) = number / = number Percentage is greater than 00: 0 % of 6 is what number? 80.6 % of 07 is what number? (0) (.0) (6) = number (80.6) (.0) (07) = number 9. = number 0.9 = number Valencia College All Rights Reserved of 0 // : AM

26 Percentage Applications Per cent age means: Out of 00 or.0 or / 00 Number % of Number = Number This number represents the original amount. Percentage amount Words refer to the same information Amount Eample : Finding the amount Eample : Eample : John owns a citrus grove with 00 trees. If % of his trees are grapefruit, how many grapefruit trees does John have? % of 00 citrus trees are grapefruit trees. % of 00 = g Reminder: The word "of" means () (.0) (00) = g that we should be multiplying. 0 = g Therefore John must have 0 grapefruit trees in this grove. Finding the percentage amount Mary has a plant nursery. If 00 of the 000 plants she grows are trees, what percent of her nursery are trees? What percent of 000 plants are 00 trees? n % of 000 = 00 n (.0) (000) = 00 n (0) = 00 n = 0 (We divided both sides by 0) Therefore 0% of the nursery was trees. Finding the original amount In the last election 0% of the people eligible to vote voted. If 00 people voted, how many people were eligible to vote? 0% of the eligible voters are the 00 people who voted. 0% of v = 00 (0) (.0) v = 00 (0.) v = 00 v = 000 (We divided both sides by 0.) Therefore there were 000 people eligible to vote in the election. Valencia College All Rights Reserved 6 6 of 0 // : AM

27 Consumer Applications: Simple Interest I represents the simple Interest you will receive. P represents the Principal which is how much you have in your account. R represents the Rate which is how much you will get at the end of a year for every $00 you have in your account. T represents the amount of Time (in years) that you leave your money in the account. Eample : I = PRT What is the interest earned on $000 invested at 6% for years? Principal = $000 (This is the amount in your account.) R = 6% or.06 per year T = years I = P R T I = (000)(.06)() I = 0 The interest earned is $0 over a -year period. Eample : What is the interest earned on $00 invested at 9% for 8 months? Principal = $00 (This is the amount in your account.) R = 9% or.09 per year. T = 8 / of a year. (Remember time is always per year.) I = P R T I = (00)(.09)(8/) I = The interest earned is $ for a period of 8 MONTHS. Valencia College All Rights Reserved 7 7 of 0 // : AM

28 Part Developmental Math I MAT008 Pr eviously called Pr ealgebr a MAT00 NOTE: The material in this section will not specifically be on the PERT test. But if you do not know this material then the possibility of getting into a higher course is unlikely. Valencia College All Rights Reserved 8 8 of 0 // : AM

29 Test for Developmental Math I a. Place the following numbers in order from least to greatest:,, 8, 0, a., 8, 0,, b.,, 0,, 8 c. 8,, 0,, d. 0, 8,,, b. Place the following numbers in order from least to greatest:.,.0,.0 a..,.0,.0 b..0,.0,. c..0,.,.0 d..,.0,.0 c. Place the following numbers in order from least to greatest:,,, a. b. c. d.,,,,,,,,,,,, a. Simplify: + ( ) ( 6) a. b. c. 8 d. b. Simplify: a. b. c. d. 8 Valencia College All Rights Reserved 9 9 of 0 // : AM

30 c. Simplify: ( 7) a. b. c. 9 d. 99 a. Simplify: a. 7 b. c. 8 d. + b. Simplify: + 7y y + a b c y d y a. Simplify: ( ) a. b. c. + d. 8 b. Simplify: ( + 6) ( + ) a b c. + d. + 0 c. Simplify: a. + b. + c. 0 + d. Valencia College All Rights Reserved 0 0 of 0 // : AM

31 a. Evaluate the algebraic epression when a =, b =, and c = 7: a + b c a. b. c. 0 d. 6a. Simplify: y ( ) + 7( y) a. y y + b. y + 7 8y c. 7 y 8y d. y + 6 6b. Simplify: (m + m 7) (m m + 9) a. m + 7m 6 b. m + m + c. 7m m +6 d. 7m + 7m 6 7a. Solve: 8( ) = ( + ) a. = b. = c. = d. = 7b. Solve: ( + ) = ( + ) a. = b. = 7 c. = d. = Valencia College All Rights Reserved of 0 // : AM

32 Developmental Math I Test Answers: a. c b. a c. b a. c b. a c. b a. c b. d a. c b. a c. c a. d 6a. c 6b. a 7a. b 7b. a Valencia College All Rights Reserved of 0 // : AM

33 DM-I Question #: Putting Numbers in Order by Relative Size Comparing fractions: Must have common denominators. Put in proper inequality: 6 Rewrite with common denominator: 9 0 Because 9 is smaller than 0: 9 < 0 Comparing decimals: Must look at common columns. Put in proper inequality:.6.68 Look at each common column individually starting from the left. Tens column: Both are equal (). Ones column: Both are equal (). Tenths column: Both are equal (6). Hundredths column: hundredths is greater than hundredths. Thousandths column: Since we have already found a larger column that determines which is greater, this is unimportant! Therefore:.6 >.68 To compare more than fractions or decimals you can use the same procedure as above to find the largest or smallest value. Valencia College All Rights Reserved of 0 // : AM

34 DM-I Question #: Operations with Integers Integers are positive and negative whole numbers including zero. Set of integers: {, -, -, -, -, -, 0,,,,,, } Integers do NOT include fractions or decimals. Addition: Positive integers: How much money you have! Negative integers: How much money you owe! Addition is the process of combining what you have and owe. Eample Read as follows Read answer Answer You owe $6 and have $8 Have $ + (-7) You have $ and owe $7 Owe $ (-) You owe $ and owe $ Owe $ You have $6 and have $ Have $0 0 Subtraction: Are you having trouble doing a subtraction problem? If so, then do the opposite of what follows the subtraction sign and work the problem as an addition problem. Eample Written as addition Answer 9 + (- 9) - 7 (-) (- 8) - - (-7) Multiplication and Division: LIKE signs give a positive answer. UNLIKE signs give a negative answer. Eample Answer Eample Answer 0 Like signs Like signs -6 (-) Unlike signs 8 (-) Unlike signs -9 - Valencia College All Rights Reserved of 0 // : AM

35 DM-I Question #: Adding and Subtracting Polynomials Terms When adding or subtracting polynomial terms, the terms must be like terms. Like terms are defined as algebraic terms which have the same variable(s) with the same eponents. Simplify: Eample Use commutative property to rearrange. -6 and 7 both have as the like term - and - both have as the like term Which are like terms? 8 Total up the terms: = Total up the terms: - = -8 Remember that like terms must have the same variable(s) with the same eponents. The following are like terms because of this fact. and -6 are like terms. 6y and y are like terms. - and are like terms. The following are not like terms: and 8y 6 y and 8 y Because they have different variables. Because the eponents are not the same on the y variable. 8 and 7 Because the eponents are not the same on the variable. Simplify: Answers: y + 8 y 8 y y Valencia College All Rights Reserved of 0 // : AM

36 DM-I Question #: Multiplying Polynomials Monomial (one term) times Binomial (two terms): Simplify: ( + 7) Eample + 7 Distributive property 0 + Multiplication of factors Simplify: ( ) (-y) Eample (-y)() (-y)() Distributive property -0y + y Multiplication of factors Binomial ( terms) times Binomial ( terms):. Multiply First terms. Multiply Outside terms Simplify: ( + 7) ( + ). Multiply Inside terms. Multiply Last terms Distributive property First Outside Inside Last Spells FOIL Did you notice that the first letter of each word F-O-I-L spells FOIL??? This is a trick to help you remember the steps to multiplying binomials! Multiplication of factors Addition of like terms Valencia College All Rights Reserved 6 6 of 0 // : AM

37 Binomial ( terms) times Binomial ( terms) continued: These problems use the same method as on the previous page. But they LOOK different because there is no middle term! Simplify: ( + ) ( ) Eample + Distributive property Multiplication of factors Combine like terms Note: This answer is called: Difference of squares because it is subtraction and both terms are perfect squares. Simplify: ( + ) MEANS ( + ) ( + ) Distribute: Answers:. ( +) ( ) 6. -( + ) -0. -( ) ( )(-6) ( + ) (7 ) y( + m + 6) 0y +my +0y 9. (- + r 7)(-) 0r (y + ) + 0y + The is a separate term and not part of the distributive property!. ( + ) ( + ) ( + ) (- + ) ( + ) (m + ) 8m m +. ( ) ( ) ( + 8) ( ) ( ) ( + ) (y + ) (y ) y 6 8. ( ) (7m + ) 9m + m + 9 Valencia College All Rights Reserved 7 7 of 0 // : AM

38 DM-I Question #: Evaluating an Epression Evaluating an epression is done by replacing a variable with a given value and then simplifying using order of operations. Evaluate: + 7 when = () + 7 Substitute for the variable + 7 Simplify using order of operations Simplify using order of operations Evaluate: 7 + yz when =, y = 7, and z = 9 7() + (7)(9) Substitute for, 7 for y, and 9 for z 7() + (7)(9) Simplify using order of operations Simplify using order of operations 9 Simplify using order of operations Evaluate: y when = 6 and y = - (6) (-) Substitute 6 for and for y + Simplify using order of operations Simplify using order of operations Evaluate: - y z when = -, y = 7, and z = -9 -(-) (-)(7) (-9) Substitute for, 7 for y, 9 for z -(-) (-)(7) 8 Simplify using order of operations Simplify using order of operations - Simplify using order of operations Evaluate Answers. y when = and y = 6 9. g p when g = and p = y z when y = 9 and z = abc m when a = -, b =, c = -6, m = r + w when r = 8 and w = -6 Valencia College All Rights Reserved 8 8 of 0 // : AM

39 DM-I Question #6: Simplifying an Epression An epression is one or more terms that are added or subtracted. Eamples:, +, -6, + m 9r + SIMPLIFY an epression using GE( MD)( AS ) :. Work inside all Groupings. Fraction ( Parentheses ), Brackets,, Radical, Absolute value Bar. Work with Eponents.. Do Multiplication and Division combinations from Left to Right.. Do Addition and Subtraction combinations from Left to Right. Simplify: Multiply Add / Subtract like terms left to right Simplify: ( + 7 ) + () (- + 7) + () Work inside parentheses (- + 7) + 6 Work with eponents Distributive property (multiplying) Note: Instructors epect terms with eponents to be written in descending order like this: 6 + Simplify: (r + r ) (r 7r + 8) r + r r + 7r 8 Distributive property Note: Subtracting will change the sign of each term being subtracted. r + 0r 0 Add / Subtract like terms. Simplify Answer ( + ) ( 8) +. ( 7) + ( ) + 8. h + (h 7) (6h) + 0 h. -(w ) (w 8w + 9) -w w + 6. (r + 6 ) + 8(-r + r + ) -6r + 6 Valencia College All Rights Reserved 9 9 of 0 // : AM

40 DM-I Question #7: Solving an Equation Equation ( epressions that are equal to each other): + 8 = Left epression = Right epression SOLVE: Means to isolate a specific variable in an equation on either side by doing the same thing to both the simplified epression on the left side and the right side.. Simplify: GE( MD)( AS ) the epressions on either side of the equation.. Collect all variables into ONE TERM (add or subtract to both sides).. Isolate the variable term (add or subtract constant term to both sides).. Isolate the variable itself (multiply or divide coefficient to both sides). Solve: ( + ) = = = = = 6 Simplify the left side Subtract constant term from both sides Simplify to isolate variable term Divide by coefficient on both sides Simplify to isolate variable Solve: ( ) = ( + ) 8 = + 0 Simplify both sides 8 8 = Subtract 8 term from both sides - = Simplify variables into one term - 0 = Subtract constant from both sides - = 7 Simplify to isolate variable term 7 Divide by coefficient on both sides 7 7 Simplify to isolate variable 7 Valencia College All Rights Reserved 0 0 of 0 // : AM

41 This page shows shortcuts that MAY be used to clear decimals or fractions from an equation before solving! Solving an equation involving a decimal: Solve:..7 =. Note: Since the most digits after any decimal point is one, if we multiply both sides by 0 we can remove the decimals. 0(..7) = (.)0 Multiply both sides by 0 7 = Distributive property Solve the equation as done with integers. Solve:. +.7 = 9 Note: Since the most digits after any decimal point is two, if we multiply both sides by 00 we can remove the decimals. 00(. +.7) = (9)00 Multiply both sides by = 900 Distributive property Solve the equation as done with integers. Solving an equation involving a fraction: Solve: 6 Note: Since the LCD for all fractions is, if we multiply both sides by we can remove the denominators by reducing. Multiply both sides by (LCD) 6 6 Distributive property Reduce fractions = 9 No denominator format Solve the equation as done with integers. Valencia College All Rights Reserved of 0 // : AM

42 WORD PROBLEMS Following these steps will help you get through a word problem:. Draw a picture that matches your information.. Put all information from the problem on your picture. (In English - Not Algebra) If you can tell what the problem says by looking at your picture, then you have done an ecellent job. OPTION: Make a chart for all your information. (Great organizational tool.). To know that you really understand the problem, try putting in a reasonable guess of the answer and then figure out if it is correct or incorrect. You could continue to guess at the answer until you get it correct, but eventually guessing will take up too much of your time.. Using algebra will allow us to find the correct answer without all the guessing. This means that we will use a variable to represent the correct answer in place of the value we were guessing. We will write an equation very similar to the work as when we were guessing. Equation clue words: TOTAL, SUM, or similar words.. Solve the algebraic equation. Check your answer. 6. Answer the original question by writing in sentence format. Eample: At a service station, the underground tank storing regular gas holds 7 gallons less than the tank storing premium gas. If the total storage capacity of the two tanks is 8 gallons, how much does the premium gas tank hold? Premium gas tank Regular gas tank TOTAL (clue word) 8 gallons Holds 7 gallons less than premium tank I am going to guess that the premium tank holds 00 gallons. To check our guess we can calculate that the regular gas tank holds gallons (7 less than the premium tank of 00 gallons). If our guess had been correct the total for the two tanks would be 8. But because 00 and totals to the incorrect value of 9, we need to guess again! Using algebra we can assign a variable (the correct answer) to the amount in the premium tank. The correct amount in the premium tank is P. Therefore the amount in the regular tank (which is 7 gallons less) is P - 7. Now since these answers are correct the total will be 8 gallons. Reasoning: Amount in premium tank + Amount in regular tank = 8 gallons Equation: P + P - 7 = 8 Solve: P -7 = 8 (Combine like terms) P = 900 (Add 7 to both sides of equation) P = 0 (Divide both sides by ) Answer: The premium tank holds 0 gallons of gas. Valencia College All Rights Reserved of 0 // : AM

43 WHAT THE WORDS MEAN Addition: Sum the sum of and + Addition: Plus plus y + y Addition: Increased by h increased by 8 h + 8 Addition: Eceeds eceeds m by 6 m + 6 Addition: Added to 7 added to m m + 7 Addition: More than h more than + h Addition: Greater than greater than y y + Subtraction: Difference the difference of k and k Subtraction: Minus minus Subtraction: Decreased by h decreased by 8 h 8 Subtraction: Reduced by reduced by r r Subtraction: Less less m m Subtraction: Less than less than c c Subtraction: Subtracted from 8 subtracted from r r 8 Multiplication: Product the product of and Multiplication: Times times m m Multiplication: Twice twice w w Multiplication: Of half of t / t Division: Per miles per gallon miles/gallon Division: Quotient the quotient of r and 7 r / 7 Division: Divided by divided by y / y Division: Ratio the ratio of z to z / Division: Split into split into n equal parts / n Eponent: Square the square of m m Eponent: Cube the cube of r r Equals: Is The sum of and r is 9 + r = 9 Equals: Result is If you increase m by 7, the result is twice m + 7 = Inequality Is not equal to y is not equal to twice y Inequality Is less than 6 is less than r 6 r Inequality Is greater than m is greater than y m y Inequality r is less than or equal to z r z Inequality p is greater than or equal to w p w Valencia College All Rights Reserved of 0 // : AM

44 GROUPING SYMBOLS Tells you to FIRST try to simplify what is inside the grouping symbol. Parentheses: (w + 7) Cannot be simplif ied inside gr ouping! The distributive property will allow you to eliminate this grouping. w + 8 Epr ession wit hout gr ouping symbol. 8r Fraction bar: Cannot be simplif ied inside gr ouping! The distributive property will allow you to eliminate this grouping. 8r Epr ession wit hout gr ouping symbol. Radical: m + 9 Cannot be simplif ied inside gr ouping! m + 9 WRONG! I NCORRECT! ERROR! + 6 Can be simplified inside square root. 6 CORRECT simplified form. Absolute Cannot be simplif ied inside gr ouping! value: WRONG! I NCORRECT! ERROR! Can be simplified inside absolute value CORRECT simplified form. NOTE: The P in PEMDAS represents Parentheses but it also applies to all other Grouping symbols. If there are parentheses inside of other parentheses, be sure to work from the INSIDE to the OUTSIDE. Valencia College All Rights Reserved of 0 // : AM

45 EXPONENTS A number in the upper right hand corner of the BASE that TELLS you how times to multiply the BASE number by ITSELF. Eample: mean = 8 (Do NOT multiply anything by.) Epand using powers of 0 for each column: / / Definition: Any number to the zero power has a value of ONE () Eamples: 0 = and 0 = Multiplying the same base with eponents: means means 8 Shortcut: Add the eponents Power of a power: ( ) means means 8 Shortcut: Multiply the eponents Eample: Eample: Eample: ( ) ( y )( y ) m (m ) ()()() ( )( )(y y ) m (m )(m ) y ( )(m m m ) 6 m Be careful: But 6 means What is the opposite of 6 squared? 6 squared is 6 and the opposite of 6 is -6. (-6) means What is negative 6 squared? (-6)(-6) is 6. Valencia College All Rights Reserved of 0 // : AM

46 EQUATION PARTS with instructions Factor (number or variable(s) that are multiplied): Term (numbers and/or variables that are added or subtracted): + 8 Epression (One or more terms): + 8 SIMPLIFY an epression using GE( MD)( AS ).. Work inside all Groupings. ( Parentheses ), Brackets, Fraction, Bar Radical, Absolute value. Eponents. Do Multiplication and Division combinations Left to Right.. Do Addition and Subtraction combinations Left to Right. Equation ( epressions that are equal to each other): + 8 = Left epression = Right epression SOLVE an equation by doing the same thing to both epressions. One is on the left side of equals sign and one is on the right side.. Simplify: GE( MD)( AS ) both sides of the equation.. Collect all variables into ONE TERM (add / subtract to both sides).. Isolate the variable term (add / subtract constant term to both sides).. Isolate the variable itself (multiply/divide coefficient to both sides). Valencia College All Rights Reserved 6 6 of 0 // : AM

47 Additional Developmental Math I Practice:. What is the perimeter of a rectangle that has a length of 8 cm and a width of cm? a) 0 sq. cm b) 0 cm c) 6 cm d) 6 sq. cm. What is the area of a square that has a side length of centimeters? a) 8 cm b) 8 sq. cm c) sq. cm d) cm. What is the area of a triangle that has a base of inches and a height of 8 inches? a) 0 inches b) 0 inches c) 0 sq. inches d) 0 sq. inches. What is the volume of a cube that measures meters on each side? a) meters b) meters c) cu. meters d) sq. meters. Use order of operations to simplify: a) 997 b) 7 c) d) 6. Use order of operations to simplify: a) b) -7 c) d) -7 (8 ) 6 ( ) 7. Use order of operations to simplify: 0 [6 ( 8)] a) b) c) - d) 9 8. Simplify: 9 ( ) a) - b) - c) d) 9. Simplify: 6 a) - b) - c) 9 d) 0. Simplify: 9 y y y 6y a) 8 y 0y b) y c) 8 y 0 y d) 8 y 0 y. Simplify: a) z z 0 z b) z c) 0 z d) 0 z Valencia College All Rights Reserved 7 7 of 0 // : AM

48 . Simplify: (.6 y)(0. z ) a).8yz b) d) 0.68yz c).68yz d).8 yz. Evaluate the following epression for = and y = -: a) - b) c) -7 d) y. Evaluate the following epression for = / and y = / : ( y)( y ) a) 7 / 6 b) / c) / d) / 9. Evaluate the following epression for = 0. and y =.: a). b). c) 0.6 d).0 ( y ) 6. Solve for : 0 a) = / b) = c) = -0 d) = - 7. Solve for : 7 a) = / b) = - / c) = - d) = 8. Solve for : (8 ) 6( ) a) = / b) = 6 / c) = 6 / d) = / 9. Solve for y: y a) y = 0 / 9 b) y = - c) y = / d) y = 9 / 0. Solve for z: 6 z 7 a) z = - / b) z = 6 / c) z = / d) z = 0 /. Solve for z:.6 0.0z 7.8 a) 0.96 b) 69 c) 0.06 d) Solve for :.8 a).7 b).88 c) 7. d) 0.6 Valencia College All Rights Reserved 8 8 of 0 // : AM

49 . The formula for the perimeter of a rectangle is: P = L + W Solve for W when P = and L = 7. a) W =. b) W = c) W = 8. d) W = 8. Best Buy is selling a television for $0.00. Sales ta in Orange County is 6%. Using P as the amount I will have to pay for the television (including sales ta), write an algebraic equation that describes this transaction. a) P = (0.06)(0) b) P = c) P = 0 + (0.06)(0) d) P = 0(0.9). Jeremy put $0 into his savings account, which pays % per year simple interest and left it there for years. Using A as the total amount that will be in the bank at the end of the years, write an algebraic equation that describes this transaction. a) A = 0 + 0(0.0)() b) A = 0(0.0)() c) A = 0 0(0.0)() d) A = 0 + (0.0)() 6. Keisha is investing her money in an IRA. Initially she will be putting in $77. Using C as the additional amount invested each month, translate this problem into an algebraic epression that will show how much Keisha invested for the entire year. a) 77 C b) 77 C c) 77C d) 77 C 7. Multiply and simplify where possible: ( ) a) 0 b) 0 8 c) d) Multiply and simplify where possible: (y z ) a) y + 0z b) -yz c) 8y + 9z 6 d) y + 0z 9. Multiply and simplify where possible: a) c) 6 y y z b) 6 y y z d) 7 y y z y z y(6 y z ) 0. Multiply and simplify where possible: ( )(6 7) a) 0 0 b) c) + d) 6 + Valencia College All Rights Reserved 9 9 of 0 // : AM

50 . Multiply and simplify where possible: a) z b) z c) z d) 9 z 6 z z. Multiply and simplify where possible: ( z)( 7 z ) a) 0 7z + 7z b) 0 + 7z c) 7 8y d) 0 7z + 7z. Simplify: (6 7) ( ) a) 9 b) 9 c) d) 9 +. Simplify: ( z z ) (7z 8z ) a) -6z z + 6 b) -6z + z c) -6z z d) -7z + z. Simplify: ( z ) (z 7) (z z 9) a) -z + z + b) -z + z + c) -z + z d) -6z + z Answers:. C. C. C. C. D 6. D 7. C 8. D 9. B 0. D. A. B. C. A. B 6. D 7. D 8. B 9. C 0. D. D. D. A. C. A 6. D 7. D 8. D 9. A 0. D. D. D. A. B. B Valencia College All Rights Reserved 0 0 of 0 // : AM

51 Part Developmental Math II MAT008 Pr eviously called Beginning Algebr a MAT00 Note: Material from this section will be on the PERT test. If you do not know the material from this section then you will probably be placed in the Developmental Math I (MAT008) course. Valencia College All Rights Reserved of 0 // : AM

52 Test for Developmental Math II a. Simplify: 8a a. 6 a b. 8a c. 8a d. 6a b. Simplify: a b. + 6 c. 6 d. 6 a. Identify the greatest common factor: y 6y a. y b. y c. d. -y b. m n factors into: a. mn(m n) b. (m n) (m n) c. (m + n) (m n) d. not factorable c. Identify a factor of the following trinomial: a. ( + ) b. ( + ) c. ( + ) d. ( 6) Valencia College All Rights Reserved of 0 // : AM

53 d. Simplify: a. b. c. d. a. Simplify: y a. y y b. 6 c. d. y b. Simplify: a. 6 b. 6 0 c. y y y 0 y d. 0 6 Valencia College All Rights Reserved of 0 // : AM

54 a. Find the and y intercepts and the slope of the given equation: + y = a. (6, 0), (0, ), b. (, 0), (0, 6), c. (6, 0 ), (0, ), d. (, 0), (0, 6), b. Match the following equation to the appropriate graph: y = + Each mark on the aes represents one () unit. a. b. c. d. a. Solve for : a. b. 8 c. d. Valencia College All Rights Reserved of 0 // : AM

55 b. Solve for : a by c a. a c by b. by c a c. c by a d. c by a c. Solve for : 0 a. b. c. d. 7, 7,, 7 7, d. Solve for : 8 a. b. c. d. 6a. Convert to scientific notation: a.. 0 b c d b. Convert to standard form:. 0 a b. 000 c d..000 Valencia College All Rights Reserved of 0 // : AM

56 Developmental Math II Test Answers: a. d b. b a. b b. c c. b d. c a. a b. a a. a b. a a. c b. c c. b d. d 6a. b 6b. b Valencia College All Rights Reserved 6 6 of 0 // : AM

57 DM-II Question #: Square Roots Your need to know from memory all the following perfect squares:, 9, 6,, 6, 9, 6, 8, 00,,,... and a, b, c, d,..., y, z Simplifying a square root means to find all perfect square factors that are located inside the square root symbol (radical) and remove them. Simplify: 8 Eample 9 Factor out the perfect square 9 9 Separate the square root of 9 from the square root of Replace the square root of 9 with the value Simplify: 7 Eample Factor out the perfect squares and Separate the perfect squares of and Replace the square roots of and with and Simplify: 6 8 y 6 Eample 6 y y y Factor out the perfect squares: 6,, y, y, y y y y Separate the perfect squares of 6,, y, y, y yyy Replace the square roots of 6,, y with,, y y Simplify the y s using eponents Like terms have the same value inside the square root (radical): Like radicals Unlike radicals and and and 7 7 and 7 m and 7 m 6 r and 7 h Eample: simplifies to: 7 Eample: 6 simplifies to: 6 Valencia College All Rights Reserved 7 7 of 0 // : AM

58 Simplify: Eample 9 Factor out the perfect squares: 9 and 9 Separate the perfect squares: 9 and Replace the square roots of 9 and with and 8 Because these are like terms they can be added Eample: 6 ( 7) Because we have unlike terms inside the parentheses we need to distribute to simplify this eample. 6 Multiplication (and division) allows us to combine any square roots. 9 6 Correct Watch out! Summary: Simplify m A B A B This is not correct. Answer m 8r r r 8 80k 6k 7 g 0g z g 00 z 6. y 7d y z d y yz k 7 6k k ( 7 ) 6. (6 7 9 ) 6 0. r ( r 8 h) r 6 0hr. 8( 8 8) 96 Valencia College All Rights Reserved 8 8 of 0 // : AM

59 DM-II Question #: Factoring A FACTOR is a number, letter, term, or polynomial that is multiplied. Factoring requires that you put (parentheses) into your epression. Step : Look for factor(s) that are common to ALL terms. Common factors are written on the outside of the parentheses. Inside the parentheses is what is left after removing the common factor(s) from each term. Eample: Factor completely the following polynomial. + Binomial epression + and are common factors ( + ) Completely factored polynomial Completely means there are NO MORE common factors. Note: Factor completely also means Find Greatest Common Factor Net we should look for factors that are binomials: Step : Step : ( )( ) ( )( ) * * Tr y f act or s t hat ar e closest t oget her in value f or your f ir st t r y. Step : The last sign tells you to add or subtract the Inside Outside terms = middle term. ( ) ( ) If your answer equals the middle term, your information is correct. Note: If there is no middle term, then it has a value of zero (0). Step : Assign positive or negative values to the Inside term () and the outside term () so the combined value will equal the middle term. Put these signs into the appropriate binomial. Valencia College All Rights Reserved 9 9 of 0 // : AM

60 DM-II Questions #: Factoring Eamples Factor completely: This means that you are epected to LOOK for any common factors (GCF) BEFORE looking for possible binomial factors. Factoring problems may have either or both of these types. Directions: Factor completely: Eample: Trinomial Step : No common factors Always look for common factor(s) first. Step : ( ) ( ) What factors equal 0 (F in FOIL)? Try factors that are closest together in value first!!!! 0 and are possibilities, but are not used as often. Step : ( ) ( ) What factors equal 6 (L in FOIL)? Since and are closer in value than 6 and we have made a good choice. But the and together have a common factor (we did common factors in step so this should not occur), therefore the factors should be switched so there are no common factor(s) in either binomial. Step : ( ) ( ) Better choice to factor this polynomial. 0 6 Outside term () and Inside term (). Are these Outside () and Inside () terms correct? YES!!!! The second sign (subtraction) indicates that if we subtract our Outside term () and Inside term () to total our factors would be correct. Since = we know our factors were placed correctly! Step : Assign appropriate signs: The combined total of the Outside () and Inside () terms will be a POSITIVE. This would require the be positive and be negative. Factored completely: ( ) ( + ) Eample: 6y 66y + 8y Factor completely: Step : 6y(6y y + ) Factor out 6y (GCF). Step : 6y(y )(y ) Factor 6y (F in FOIL). Step : 6y(y )(y ) Factor (L in FOIL). Step : Outside term (9y) + Inside term (y) = y This is CORRECT!! Step : Assign appropriate signs: The combined total of the Outside (9y) and Inside (y) terms will be a NEGATIVE. This would require both the 9y and y be negative. Factored completely: 6y (y ) (y ) Valencia College All Rights Reserved of 0 // : AM

61 DM-II Question #: Reducing (Simplifying) Rational Epressions STEPS FOR SIMPLIFYING:. Put parentheses around any fraction bar grouping to remind you it is ALL or NOTHING. Factor out all COMMON factors. Factor out all BINOMIAL factors. Find all restrictions (Values of the variable that will cause the denominator to equal zero). Reduce (Watch for GROUPING symbols) Eample: + 6 _ Step : ( + 6) This grouping has common factors of and ( + + 6) This grouping has binomial factors of ( + ) and ( + ) Step &: ( + ) _ ( + )( + ) Step : Restrictions: { -, -} [Remember that you cannot divide by zero.] Step : ( + ) ( + ) is a binomial factor of the numerator and ( + )( + ) the denominator. REDUCE to. Simplified: _ ( + ) Questions. Answers. a 0 a. 8k k 6 8 k. 7 Valencia College All Rights Reserved 6 6 of 0 // : AM

62 DM-II Question #: Eponent Shortcuts When simplifying rational epressions (fractions) with variable factors, the object is to have the variable appear only once in the epression with a positive eponent.. a b a b ( )( ) When multiplying with same base, add eponents. Eamples: 6 9 ( )( ) ( r )( r ) r ( p )( p )( p ) 7 p. a b a b When dividing with the same base, subtract the eponents. Eamples: r r 9 r or w w 8 w or h h h 8. a b ab ( ) When raising a power to a power, multiply the eponents. Eamples: ( g ) g ( ) 0 a. a or a a To remove a negative eponent, you will need to find the reciprocal of what the negative eponent is located on. Eamples: n or n v 7 v 7 9 z z z z z or 9 9 z z z z z z Valencia College All Rights Reserved 6 6 of 0 // : AM

63 DM-II Question #: GRAPHING Linear Equations General Information: Vertical ais (Output) y Quadrant II Quadrant I Horizontal ais (Input) Origin Quadrant III Quadrant IV (Start counting here to locate a point) -coordinate (Go to the right 7 units from the origin) y-coordinate (Go down units from the origin) (7, ) Ordered pair (-value, y-value) A point (represented by an ordered pair) is located by starting at the ORIGIN (where the aes cross) then move horizontally according to the -coordinate and then vertically according to the y-coordinate. It is assumed that you will count by ONES on a graph, if not, you MUST note your scale on the aes. The two aes do NOT have to have the same scale. Graphing a linear equation: [ and y are to the st power] The graph will always be a straight line with arrows on both ends. Eample: y = Table of values y Most values are found by picking a value for (that will fi on your graph paper) then put this value into your equation to calculate the y-value. 0 For best results attempt to use positive, negative, and zero values. Valencia College All Rights Reserved 6 6 of 0 // : AM

64 Linear equations formats: Slope-intercept: y = m + b Point-slope: y y = m( ) Point-slope: y = m( h) + k Graphing information: Point = (, y ) y - intercept ( = 0) - intercept (y = 0) Y (answer) in terms of X (work) Output in terms of Input Graphs Slope: You can find the rise and run from any points on your graph! Run y-intercept ( = 0) Rise Input Output -ais y-ais -intercept (y = 0) To put a line on your graph paper: Step : Establish a point on your graph. Step : Use the slope to establish a second point. Step : Draw your line through the points. Valencia College All Rights Reserved 6 6 of 0 // : AM

65 DM-II Question #: Linear Graph, Slope, -intercept, y-intercept A linear equation can give specific information (slope and y-intercept) about a graph when written in the slope-intercept format. Slope Intercept form of a linear equation: y = m + b This is a positive slope (m > 0) because you see the graph rise as you read from left to right. This is a negative slope (m < 0) because you see the graph fall as you read from left to right. y intercept (b) is the point where your graph crosses the y-ais. Any form of a linear equation: To find -intercept: Set y = 0 (or cover up the y-term) and solve for. To find y-intercept: Set = 0 (or cover up the -term) and solve for y. To find slope: Rewrite in Slope Intercept format and read m-value. Question Answer. y = + 7 Find slope & y-intercept Slope = ; y-intercept = 7. y = - Find slope & y-intercept Slope = - ; y-intercept = -. y = / + 8 Find slope & y-intercept Slope = / ; y-intercept = 8. + y = Find slope & y-intercept Slope = - / ; y-intercept = /. + y = Find and y-intercepts ( /,0) and (0, / ) 6. y = 0 Find and y-intercepts (, 0) and (0, -) y = Find the slope Slope = 6 / Valencia College All Rights Reserved 6 6 of 0 // : AM

66 DM-II Question #: Solving Linear Equation: [One variable Answer is a number]. Simplify both sides of the equation individually. This means to apply GEMDAS to both sides.. Collect the variable you are solving from both sides into term. This may be done by adding or subtracting the variable term to both sides.. Isolate the term containing the variable you are solving for. This may be done by adding or subtracting the constant term (the one without a variable in it) to both sides.. Isolate the variable you are solving for. This may be done by dividing by the coefficient on both sides. Literal Equation: [Multiple variables Answer has variables in it] Similar to Linear Equation above, but uses more than one variable. Inequality: [One variable Answer is a number]. Simplify both sides of the equation individually. This means to apply GEMDAS to both sides.. Collect the variable you are solving from both sides into term. This may be done by adding or subtracting the variable term to both sides.. Isolate the term containing the variable you are solving for. This may be done by adding or subtracting the constant term (the one without a variable in it) to both sides.. Isolate the variable you are solving for. This may be done by dividing by the coefficient on both sides. If YOU multiplied or divided both sides by a negative number, you must reverse the inequality. Eample: + 7( ) > + 8 Step : + 7 > > + 6 Step : 9 > > 6 Step : - + > > 0 Step : 0 < < - *Divided by a negative no.* *Reverse the inequality* Valencia College All Rights Reserved of 0 // : AM

67 Quadratic Equation: [ Has and two answers] General form: a + b + c = 0 Square root method: Use if there is no term [b = 0] Eample: 00 = 0 = 00 (Add 00 to both sides) = 00 = = (Divide both sides by ) (Simplify each side) (Take square root of both sides) = (Simplify each side) Answers: = {-, } Factoring method: Set equation = 0 [General form] Eample: + = 0 [c 0)] ( + 6)( ) = = 0 or = 0 = -6 = Answers: = {-6, } Eample: + = 0 [c = 0] ( + ) = 0 = 0 or + = 0 = 0 = - Answers: = {-, 0} Quadratic formula method: Set equation = 0 [General form] E: + 7 = 0 [a =, b =, c = -7] = = = = b b a ac 6 8 ( 7) Valencia College All Rights Reserved 67 = = 9 9 therefore: and = = and = 7 67 of 0 // : AM 9

68 DM-II Question #6: Scientific Notation Definition of scientific notation: A between number and 0 multiplied by a power of 0. Eample:.07 0 or Logic: If we multiply a number by 0 one or more times, the value of the number will get larger. Rule: Positive powers of 0 will move the decimal point to the RIGHT! Eample: 7. 0 Eample: Logic: If we divide a number by 0 one or more times, the value of the number will get smaller. Rule: Negative powers of 0 (same as dividing by 0) will move the decimal point to the LEFT. Eample: Eample: Questions Answers Put the following scientific numbers into standard form:.. 0, ,908,000, ,77,000,000, Put the following standard form numbers into scientific notation: 6.,000, ,800,000,000, Valencia College All Rights Reserved of 0 // : AM

69 Setting up Word Problems - Using Variable. Draw a picture that matches your information.. Put all information from the problem on your picture. Option: Make a chart for all your information. (Great organizational tool.). Look for relationships among the information. (Definition, formula, logical conclusion). If lost, try assigning a reasonable guess to the question and calculate if the guess is correct or incorrect. Guessing will give a better understanding of the information.. Assign a variable (the correct value) to the unknown and write an equation that should be similar to the information that you were guessing in the previous step. 6. Solve equation. 7. Answer the original question in sentence form. Eample: The length of a rectangle is feet more than the width. The perimeter of the rectangle is 88 feet. Find the length of the rectangle. Length = feet more than the width (W + ) Width Perimeter: Distance all the way around the outside of the rectangle is 88 feet. Definition: Perimeter is equal to the sum of the four sides. Variable: If I knew the width, then I could use that info to find the length. Therefore I will assign the variable W to represent the correct width and W + to represent the correct length. Setup: Perimeter = Width + Length + Width + Length 88 = W + (W + ) + W + (W + ) Eample: Two cars leave the same point at the same time traveling in opposite directions. One car travels west at 0 miles per hour and the other travels east at 60 miles per hour. In how many hours will they be 80 miles apart? Goes west at 0 mph Goes east at 60 mph Distance apart is 80 miles Definition: Distance traveled is equal to the product of the rate of speed and time elapsed.(d=r T) Logic: The distance apart is the total mileage of both cars because they are going in opposite directions. We also total the rates for the same reason. But travel time is unaffected by direction. Time in this problem will be the same for both cars because they start and stop at the same time. Variable: Use T as the correct number of hours of travel time for the cars. Setup: Distance = Rate of speed Time 80 miles apart = 80 miles apart after each hour (80 mph) T hours traveling Eample: Find the amount of money now necessary to be invested at % simple interest to yield $0 interest in years. (Yield means the amount of money produced at end of time in bank.) Principal ($ you put in bank) % Rate of interest years later = $0 total interest Definition: Principal times Rate of interest per year times Number of years equals Total interest. Variable: P is the correct amount of money we need to put into the bank. Setup: Principal Rate Time = Interest (yield) P 0.0 = $0 Valencia College All Rights Reserved of 0 // : AM

70 Geometry Point Basic geometric figure. Segment Two points and all those in between. Part of a line. Ray Starts at a point and etends forever in one direction. Line Etends forever in opposite directions. Angle Measurement (in degrees) around a point. Formed by two rays with a common endpoint (verte). Congruent Two geometric figures with same measurement. Parallel lines Do not intersect. Have the same slope. Perpendicular lines Meet at right angles. Their slopes are opposites and reciprocals. Single angles: Acute angle: Between 0º and 90º. Right angle: 90º. Formed by perpendicular lines. (Do not assume.) Obtuse angle: Between 90º 80º. Straight angle: 80º If it looks like a straight angle, then assume it is. Pairs of angles: Complementary angles: Any two angles that total 90º. Supplementary angles: Any two angles that total 80º. Vertical angles: Opposite angles formed by two intersecting lines. Measurements of vertical angles are always equal. Triangle: A -sided polygon with angles that total 80º. Equilateral Isosceles Right triangle sides & angles sides & angles a + b = c Similar triangles Corresponding angles are congruent ( ). Corresponding sides are proportional. Valencia College All Rights Reserved of 0 // : AM

71 Perimeter (units): Area (square units): Distance around the outside of polygon. Number of squares you can count inside of a polygon. Square side Perimeter = s units Area = s square units Rectangle width Perimeter = l + w units Area = lw square units length Parallelogram height Perimeter = Add all sides together. Area = bh square units base Triangle height Perimeter = Add all sides together. Area = ½ bh square units base Rectangular height Surface area = lw + lh + wh sq. units solid width Volume = lwh cubic units length Number of cubes in the bo! Circle Circumference = r or d units Diameter (d) Area = r square units Radius (r) Pi ( ) is the number (.) of diameters in the circumference. Valencia College All Rights Reserved 7 7 of 0 // : AM

72 RATIONAL EXPRESSIONS Multiplication/Division STEPS FOR MULTIPLYING (or dividing) FRACTIONS:. Put parentheses around any fraction bar grouping to remind you it is ALL or NOTHING.. Factor out all COMMON factors.. Factor out all BINOMIAL factors.. Find all restrictions (Values of the variable that will cause the denominator to equal zero).. If division: Find the reciprocal of the fraction AFTER the division sign and put in mult. sign. 6. Multiply your numerators and then multiply your denominators. NOTE: Do NOT actually multiply groupings, but show to be multiplied. 7. Reduce. Multiplication eample: Division eample: Step : ( ( ) 8) ( ) ( ( ) 8) (6 8) Step &: ( ( ) ) ( )( ) ( ( ) ) 6( )( ) Step : Restrictions: {-, -, } Restrictions: {-, 0, } Step : ( ( ) ) 6( )( ) Step 6: ( ( )( ) )( ) 6( )( ( )( ) ) Step 7: The factors and (X + ) are The factors 6, X, and (X - ) are common to the numerator and common to the numerator and denominator. REDUCE to. denominator. REDUCE to. Simplified: ( 6 )( ) ( )( ) Valencia College All Rights Reserved 7 7 of 0 // : AM

73 RATIONAL EXPRESSIONS Addition/Subtraction STEPS FOR Adding or Subtracting FRACTIONS:. Put parentheses around any fraction bar grouping to remind you it is ALL or NOTHING.. Factor out all COMMON factors.. Factor out all BINOMIAL factors.. Find all restrictions (Values of the variable that will cause the denominator to equal zero).. Find a COMMON denominator (If you already have one go to step 6). A. Write all factors of FIRST denominator. B. Multiply by ANY OTHER factors of second denominator. C. Multiply by ANY OTHER factors of subsequent denominators. D. Use identity of multiplication to get your fractions to this common denominator. 6. Add / Subtract your numerators. (Keep your same common denominator). (Remember that subtraction will CHANGE the sign of EVERY TERM being subtracted). 7. Simplify NUMERATOR (NOT your denominator). 8. Factor numerator. 9. Reduce. Eample: 6 9 Step : ( ) (6 9) ( ) Step &: ( ) ( ) ( ) Step : Restrictions: {0, } Step A,B,C: Lowest Common Denominator: ( ) Step D: ( ( ) ) ( ) ( ( ) ) Step D: ( ( ) ) ( ) ( ( ) ) (Simplified by multiplication) Step 6: ( ) ( ( ) ) Step 7: ( ) Step 8: Step 9: ( )( ) ( ) Valencia College All Rights Reserved 7 7 of 0 // : AM

74 Signed Numbers Review Simplify:. ( 8) ( 8) 6. ( ) 7. ( )(6) 8. ( 6)( 8) 9. ( 8) 0. 9 ( ) 6. 6 ( 9). ( )... ( ) 6. ( ) (.) (.07) 9. (.)(.0) 0. (.)(.). 9.0(.008) ( 0.09). ( 0.8). ( 0.) ( )(0.8) 6..6.( 0.9) 7. (. )( 0. ) 8. (.6) ( ) ( 0.) (.) Valencia College All Rights Reserved 7 7 of 0 // : AM

75 Answers: or Valencia College All Rights Reserved 7 7 of 0 // : AM

76 Linear Equation Review Solve:. ( ) 8.. y z y z z. y ( ) y y y 0.y 8. 6z 9 8 z ( ) z z. y z 9 ( ) z z ( ) 7. 6(y ) ( y ) (0. z z ) 0. ( y ) y 9y. ( 7) (9 )... 6.z..(0.8z ) Valencia College All Rights Reserved of 0 // : AM

77 Answers:... y z y z y y y z z 6. y... z y z.8 0. y z Valencia College All Rights Reserved of 0 // : AM

78 Graphing Review. Mark and label the following points on the graph below. A = (,); B = (-,-6); C = (7,-); D = (0,) Draw and label both aes. Show appropriate scale.. Fill in the chart with appropriate values that satisfy this equation: y = y Fill in the chart with 6 appropriate integer values that satisfy this graph: Units are equal to one. y Valencia College All Rights Reserved of 0 // : AM

79 . Find the and y - intercept: (Write each as an ordered pair.) + y = intercept: y intercept: y = 0 intercept: y intercept: - y = 7 intercept: y intercept: y = intercept: y intercept: = 6 intercept: y intercept:. A. Graph below: Slope = / y intercept = B. Graph below: Slope = - y intercept = 0 C. Graph below: Slope = / y intercept = - 6. Graph below: = Graph below: y = - Valencia College All Rights Reserved of 0 // : AM

80 7. Find the slope of each graph: Slope = Slope = Graph Answers: All spaces will represent one unit unless otherwise noted.. Point A will be spaces to the right and spaces up from the origin. Point B will be spaces to the left and 6 spaces down from the origin. Point C will be 7 spaces to the right and spaces down from the origin. Point D will be spaces up from the origin on the y-ais.. y / y Find the and y - intercept: (Write each as an ordered pair.) + y = intercept: (6, 0) y intercept: (0, ) y = 0 intercept: (0, 0) y intercept: (0, -) - y = 7 intercept: ( -7 /, 0) y intercept: (0, -7 / ) y = intercept: ( /, 0) y intercept: (0, -) = 6 intercept: (6, 0) y intercept: none A. This graph should go through the y-ais at and through the -ais at -. B. This graph should go through the origin (0,0) and points (, -) and (-, ). C. This graph should go through the y-ais at - and through the -ais at. 6. The graph of = is a vertical line that goes through the -ais at. The graph of y = - is a horizontal line that goes through the y-ais at The slope of the first graph is -. The slope of the second graph is /. Valencia College All Rights Reserved of 0 // : AM

81 .. y 8 6y. y z z 8y y y 6. 8z z y y y 0. 6y z 9z y z. ( )(6 y ). (7 )( ). ( )(8 ). ( )(9 y)( z ). ( 7 )( ) 6. ( )( y ) 7. ( )(7 ) 8. ( z )( z )( z ) 9. ( 8 )( 6 y )( z ) 0. ( z )(6 y )( z )( y ). ( ). (8 y ). ( z ). ( ). y( y 6 z ) 6. ( )( ) 7. ( )( ) 8. ( y)( y ) Polynomial Review 9. ( )( ) 0. ( y)( ). (7y ). ( y z ). ( z ). z( y 7 z ). ( )( y ) 6. y y y z 6yz z z 0. y 6y.. z z y y 7y. (y ) ( y ). ( y 7) y( ). ( z ) ( z z ) 6. ( y) y( ) 7. Factor: 7 8. Factor: z 8z 9. Factor: Factor: Valencia College All Rights Reserved 8 8 of 0 // : AM

82 Answers:.. y. 7y z.. 6y 6. z y 0. y 7z. y... 08yz. 6. y z 9. 8 y z y z. 6. 0y 0. z 6.. y y 0yz y y y 0y. 9y 8y. y 0yz z. z 9z. 6z 8y z z. 8 6y y y y y z z y z z y 7y y 8. y 8y. z z 6. y 8y 7. ( )( ) 8. ( z )( z ) 9. ( )( ) 0. ( )( ) Valencia College All Rights Reserved 8 8 of 0 // : AM

83 Practice: Variety of topics. Simplify: a. -8 b. - c. 60 d. -0. Simplify: a. 6 b. -0 c. 86 d. 6/. Simplify: a. b. -7 c. - d. 7. Simplify: 6 0 ( ) (7 9) ( ) ( ) a. 9 b. c. 9 d.. Evaluate the given epression when = - and y = -: a. b. 6 c. - d. - y y 6. Solve for z: z 7 (z 7) z 7 z a. b. c. d. z 7 z 7 7. Solve: a. b. c. d. 8. Solve for W: a. b. W P L c. W P L d. P L W W P L W P L Valencia College All Rights Reserved 8 8 of 0 // : AM

84 9. Twice the sum of a number and 8 is the same as the difference of and the number. Find the equation that could be used to find the number,. a. 8 b. ( 8) c. ( 8) d Two cars start from the same place at the same time. One car travels west at 70 miles per hour and another travels east at 0 miles per hour. In how many hours will they be 0 miles apart? a. 0/9 hours b. hours c. /9 hours d. hours. Identify the proportion listed below that solves this problem. It will take people to pick watermelons in a day. How many people will it take to pick 00 watermelons in a day? a. b. c. 00 d.. Simplify: 7 7 ( y ) 9 y a. 9 y b. c. y d. 9 y. Simplify: y y 7 a. y 7 7 b. y c. y d. y 7 y. Simplify: t y t a. b. c. d. t t t y 6 8 Valencia College All Rights Reserved 8 8 of 0 // : AM

85 . Convert to scientific notation: a b c d. 6. Simplify: a. b c. d. (7 7) ( 9) Simplify: a. b. z c. 8 9z d. z 8. Simplify: a. 8 7 b. c. 8 7 d. 9. Factor completely: a. 6 y( y ) b. c. 6 y( y ) d. 0. Factor completely: a. b. ( y)( y) ( y) c. ( y ) d. (8 y)( y). Factor completely: (6 7 z) 6 z ( 9)(6 ) y 8 y y y(6y 8 ) y(y 6 ) 6 y 6 8 y y a. ( )( y ) b. ( )( y) c. ( )( y ) d. ( )( y). Identify a factor of this trinomial: 6 7 ( ) ( ) a. ( ) b. c. d. (6 ) Valencia College All Rights Reserved 8 8 of 0 // : AM

86 6. Simplify: 7 6 a. b. c. d.. Solve: a. 8, b. 8, c., d. 6,. Solve: 0 7 a., 7 b., 7 c., d., 7 6. Assuming the variable represents a non-negative number, simplify completely: 6 z w a. b. w z z c. 6w z z 6w z d. 7. Simplify completely: w z z a. 8 b. 6 c. d Solve: 8(7 ) 0 a. b. c. 60 d. 9. Find the -intercept for: y 0 0 a. 0, 0, 0 b. c.,0 d.,0 Valencia College All Rights Reserved of 0 // : AM

87 0. Find the graph that best matches the given linear equation: y 6 a. b. c. d. Answers:. b. a. d. c. b 6. a 7. c 8. b 9. c 0. c. a. d. c. b. a 6. a 7. d 8. b 9. a 0. a. b. c. d. b. a 6. b 7. b 8. d 9. d 0. b Valencia College All Rights Reserved of 0 // : AM

88 Part Intermediate Algebra Note: This section will also be on the PERT test. If you know all this material then your score should show that you are ready for College Algebra. Valencia College All Rights Reserved of 0 // : AM

89 Test for Intermediate Algebra a. Solve for : < < a. 7 < < b. < < c. 7 < < d. < < 7 a. Solve the system of equations: a + b = a b = a. (, ) b. (, ) c. (, ) d. (, ) b. Solve the system of equations: + y = = 8 y a. (-, ) b. (, -) c. (, ) d. (-, -) Valencia College All Rights Reserved of 0 // : AM

90 c. Solve the system of equations: 6 + 7y = y = a. (, ) b. (-, -) c. (, -) d. (-, ) a. Simplify: 9 8 a. b. c. d b. Simplify: y 0y y 9 y y y a. b. y( y y y( y y ) ) c. d. y y y y y Valencia College All Rights Reserved of 0 // : AM

91 c. Add and simplify: a. b. c. d. ( ) ( ) 9 d. Simplify: a. ( )( ) b. c. ( ) ( )( ) ( )( ) ( ) ( ) d. ( )( ) e. Simplify: a. ( )( ) b. ( )( ) c. ( )( ) d. ( )( )( ) Valencia College All Rights Reserved 9 9 of 0 // : AM

92 f. Solve: 9 a. b. 7 7 c. 7 d. 7 a. f() = + find f(-) a. 7 b. c. d. b. f() = - + find if f() = -9 a. b. c. d. c. Given f() =, find f(a + ) a. a a + b. a + a + c. a a + d. a a + Valencia College All Rights Reserved 9 9 of 0 // : AM

93 a. Simplify (put in rationalized form): a. b. c. d. b. Simplify (put in rationalized form): a. 8 b. c. d a. Write a linear equation in standard form that crosses (, -) and (-, 9): a. y = - b. y = c. + y = d. y = Valencia College All Rights Reserved 9 9 of 0 // : AM

94 6b. Write a linear equation in slope-intercept form that is perpendicular to y = crossing (, 6): a. y 7 b. y 7 c. y d. y Intermediate Algebra Test Answers: a. a a. b b. a c. d a. c b. b c. a d. d e. c f. a a. b b. b c. b a. a b. c 6a. c 6b. a Valencia College All Rights Reserved 9 9 of 0 // : AM

95 IA Question #: Solving Linear Inequalities Solve for : < < When solving a compound inequality we want to isolate in the middle of the inequality. Remember that we must do the same thing to all three sections of the inequality. < + < 7 Eample < + < 7 Subtract from all three sections - < < Simplify 6 Simplify Divide all three sections by Note: If we multiply or divide an inequality by a negative number you must change the direction of the inequality. < - < Eample Divide all sections by - Remember to reverse the direction of the inequality sign! > > Simplify < < Rewrite the inequality starting with smaller value. Questions Answers. < < < <. < + < < < 0. < + < < <. < + < 6 Valencia College All Rights Reserved 9 9 of 0 // : AM

96 IA Question #: SOLVE A SYSTEM OF EQUATIONS by ELIMINATION Eliminate a variable by adding the equations:. Line up the variables.. If adding the equations does not give a value of zero to one of the variables, then multiply either or both equations so that zero will be one of the totals when you add them.. Add the equations.. Solve the new equation.. Substitute your value back into either of the original equations and solve for the other variable. 6. Write your answer as an ordered pair. Eample : Variable sum = 0 Eample : + y = Given equation 8y = -6 y = 6 Given equation - + y = 6 = 8 Add the equations -y = 9 = Simplify y = - () + y = Substitution 8(-) = y = Simplify + = -6 y = 6 Simplify = -0 y = 6/ Simplify = -0 (, 6/) Answers (-0, -) Eample : Variable sum 0: Choosing what to multiply by is the same idea as finding a common denominator. Remember to make one term positive and one negative. y = -8 ( y) = (-8) 0 y = -0 + y = -6 ( + y) = (-6) 9 + y = -8 Add the equations 9 = -8 Simplify = - Substitute into either of the original equations (-) + y = -6 Simplify -6 + y = -6 Simplify y = 0 Simplify y = 0 Answer: (-, 0) Valencia College All Rights Reserved of 0 // : AM

97 IA Question #: SOLVE A SYSTEM OF EQUATIONS by SUBSTITUTION Eliminate a variable by substituting one equation into the other. LOOK for = something or y = something.. Solve either equation for one of the variables.. Substitute this equation into the other one. This will leave you with one equation with only one variable.. Solve the new equation.. Substitute your value back into either of the original equations and solve for the other variable.. Write your answer as an ordered pair. Eample : or y = something Eample : y = 8 Given equation + y = + y = Given equation = 9 + y + ( 8) = Substitution (9 + y) + y = + 8 = Simplify 6 + 0y + y = 8 = Simplify 6 + y = = Simplify y = - = Simplify y = - y = () 8 Substitution = 9 + (-) y = - Simplify = (, -) Answer (, -) Note: If you do not have an equation with = something or y = something: Option : Solve one of the equations for or y and then use substitution as shown in eamples above. Option : Line up the variables and solve by elimination as shown in eamples on the previous page. Valencia College All Rights Reserved of 0 // : AM

98 IA Question #a: RATIONAL EXPRESSIONS Simplifying a Fraction STEPS FOR SIMPLIFYING ONE FRACTION:. Put parentheses around any fraction bar grouping to remind you it is ALL or NOTHING.. Factor out all COMMON factors.. Factor out all BINOMIAL factors.. Find all restrictions (Values of the variable that will cause the denominator to equal zero).. Reduce (Watch for GROUPING symbols). Eample: Step : Step &: 6 6 This grouping has common factors of and. ( 6) ( 6) This grouping has binomial factors of ( + ) and ( + ). ( ) ( )( ) Step : Restrictions: { -, -} Step : Simplified: ( ) ( )( ) ( + ) is a binomial factor of the numerator and the denominator. REDUCE to. Questions Answers.... a 0 8k 6 k 7 a 8 k Valencia College All Rights Reserved of 0 // : AM

99 IA Question #bd: RATIONAL EXPRESSIONS Multiplication/Division STEPS FOR MULTIPLYING (or dividing) FRACTIONS:. Put parentheses around any fraction bar grouping to remind you it is ALL or NOTHING.. Factor out all COMMON factors.. Factor out all BINOMIAL factors.. Find all restrictions (Values of the variable that will cause the denominator to equal zero).. If division: Find the reciprocal of the fraction AFTER the division sign and put in mult. sign. 6. Multiply your numerators and then multiply your denominators. NOTE: Do NOT actually multiply groupings, but show to be multiplied. 7. Reduce. Multiplication eample: Division eample: Step : ( ( ) 8) ( ) ( ( ) 8) (6 8) Step &: ( ( ) ) ( )( ) ( ( ) ) 6( )( ) Step : Restrictions: {-, -, } Restrictions: {-, 0, } Step : ( ( ) ) 6( )( ) Step 6: ( ( )( ) )( ) 6( )( ( )( ) ) Step 7: The factors and ( + ) are The factors 6,, and ( - ) are common to the numerator and common to the numerator and denominator. REDUCE to. denominator. REDUCE to. Simplified: ( 6 )( ) ( )( ) Valencia College All Rights Reserved of 0 // : AM

100 IA Question #ce: RATIONAL EXPRESSIONS Addition/Subtraction STEPS FOR Adding or Subtracting FRACTIONS:. Put parentheses around any fraction bar grouping to remind yourself it s ALL or NOTHING. Factor out all COMMON factors. Factor out all BINOMIAL factors. Find all restrictions (Values of the variable that will cause the denominator to equal zero). Find a COMMON denominator (If you already have one go to step 6) A. Write all factors of FIRST denominator B. Multiply by ANY OTHER factors of second denominator C. Multiply by ANY OTHER factors of subsequent denominators D. Use identity of multiplication to get your fractions to this common denominator 6. Add/Subtract your numerators (Keep your same common denominator) (Remember that subtraction will CHANGE the sign of EVERY TERM being subtracted) 7. Simplify NUMERATOR (NOT your denominator) 8. Factor numerator 9. Reduce Eample: 6 9 Step : ( ) (6 9) ( ) Step &: ( ) ( ) ( ) Step : Restrictions: {0, } Step A,B,C: Lowest Common Denominator: ( ) Step D: ( ( ) ) ( ) ( ( ) ) Step D: ( ( ) ) ( ) ( ( ) ) (Simplified by multiplication) Step 6: ( ) ( ( ) ) Step 7: ( ) Step 8: Step 9: ( )( ) ( ) Valencia College All Rights Reserved of 0 // : AM

101 IA Question #f: Rational Equations. Find a common denominator using the following steps: A. Factor out all COMMON and BINOMIAL factors. B. Write all factors of FIRST denominator. C. Multiply by ANY OTHER factors of subsequent denominators.. Multiply both sides of the equation (EACH TERM) by this common denominator. (This step will eliminate all denominators from your equation after you simplify.). Solve equation. (Check to see if answer might be a restricted value.) Eample: Step : Lowest Common Denominator: () ( + ) Step : ( ) ( ) ( ) Step : () ( ) Simplified (Reducing to eliminate denominators) 0 Step : *** Option: IF YOU HAVE A PROPORTION: FRACTION = FRACTION *** Eample: 0 ( ) Note: As eplained above you COULD multiply BOTH sides of the equation by the LCD. Optional shortcut for proportion: Cross - Multiply to eliminate denominators. ( ( 0 )(0) )(0) ( 60 ) 60 Valencia College All Rights Reserved 0 0 of 0 // : AM

102 IA Question #: FUNCTIONS For each input of a function there is one and only one output. (In other words: Each question has one and only one answer.) Reading a function: f() is read f of g() is read g of f(m) is read f of m g() is read g of r(z ) is read r of z A specific function is identified by the variable in front of the parentheses. The input variable is: (inside the parentheses). What you do with the input variable is determined by the RULE that follows the equals sign. On a graph the function (output) is represented by the vertical or y-ais Eample: f() = + Read: f of equals +. This specific function is called f. The input variable is. Rule: +. The output is what f() equals. Whatever value you choose to input for will be put into the RULE to find the value (output) of this function. f(6) = 6 + f(-) = - + f(m ) = (m ) + f(6) = 0 f(-) = -9 f(m ) = m Eample: g(m) = m + m 7 Read: g of m equals times m squared plus times m minus 7. This specific function is called g. The input variable is m. Rule: m + m 7. The output is what g(m) equals. Whatever value you choose to input for m will be put into the RULE to find the value (output) of this function. g() = () + () 7 g(-z) = (-z) + (-z) 7 g() = () + 7 g(-z) = (6z ) z 7 g() = g(-z) = z z 7 g() = 8 Valencia College All Rights Reserved 0 0 of 0 // : AM

103 IA Question #: Rationalizing an epression A rationalized epression means: No fraction inside the radical. No radical in the denominator of the fraction. How to remove a fraction inside the radical: Eample (Not rationalized) Multiply numerator and denominator by the value of the numerator (identity factor). Multiply Separate radicals Rationalized form How to remove a radical in the denominator: Eample (Not rationalized) 6 Multiply numerator and denominator by the conjugate ( ) of the denominator. Simplify Rationalized form Valencia College All Rights Reserved 0 0 of 0 // : AM

104 IA Question #6: Writing Linear Equations Parallel lines: Have the same slopes. Perpendicular lines: Have the opposite and reciprocal slopes. Slopes of - and / represent perpendicular lines. Slopes of / 7 and - 7 / represent perpendicular lines. Slope format: y y Use when you know any points. Equation formats: Slope-Intercept form: y = m + b Know slope and y-intercept. Point-Slope form: y y = m ( ) Know slope and point (, y ) Standard form: a + by = c No fractional values for a, b, or c and value of a is positive. Find a linear equation that crosses (, -) and (-, 9) : Slope: y y 9 ( ) = = 6 Since we know the slope = - and a point (,-), use point-slope form. y (-) = - ( ) Substitute slope and point value into form. y + = Simplified (Equation is not in a specific form). y = - + Simplified to Slope-Intercept form. + y = Simplified to Standard form. Find a linear equation that is perpendicular to y = 7 and crosses (,6): Slope of given equation: Slope of a perpendicular line: - / Since we know the slope = - / and a point (,6), use point-slope form. y 6 = - / ( ) Substitute slope and point value into form. y 6 = - / + Simplified (Equation is not in a specific form). y = - / + 7 Simplified to Slope-Intercept form. (y) = (- / +7) y = y = Multiply both sides by to eliminate fraction. Distributive property Simplified to Standard form. Valencia College All Rights Reserved 0 0 of 0 // : AM

105 Setting up Word Problems - Using Variables. Draw a picture that matches your information.. Put all information from the problem on your picture. Option: Make a chart for all your information. (Great organizational tool.). Look for relationships among the information. (Definition, formula, logical conclusion). If lost, try assigning reasonable guesses to the questions and calculate if they are correct or incorrect. Guessing will give a better understanding of the information.. Assign a variable to each of the unknowns and then write two equations that should be similar to the information that you were guessing in the previous step. 6. Solve equations. 7. Answer the original question(s) in sentence form. Eample: Arctic Antifreeze is 8% alcohol and Frost No More is 0% alcohol. How many liters of each should be mied to get 0 liters of a miture that is % alcohol? 8% Arctic Antifreeze (? Liters) 0% Frost No More (? Liters) % Miture (0 Liters) Pure alcohol (8% of? Liters) Pure alcohol (0% of? liters) Pure alcohol (% of 0 Liters) Logic: The percentages represent the amount of PURE alcohol that is mied with water. The amount of pure alcohol in both original containers should total the amount in the miture. Variables: Let A be the correct amount of Liters for the Arctic and F be the correct amount for the Frost. Setup : Amount of liters of Arctic + Amount of liters of Frost = Total liters for miture A liters + F liters = 0 liters Setup : Amount of pure alcohol in Arctic + Amount of pure alcohol in Frost = Pure alcohol in miture 8% of A liters + 0% of F liters = % of 0 liters Eample: The Nutty Professor mies 0 pounds of cashews that cost $6.7 per pound and some Brazil nuts that cost $. per pound. If he wants to make a miture that costs $.70 per pound, how many pounds of Brazil nuts should he use? 0 pounds of cashews at $6.7? pounds of Brazil nuts at $.? pounds of miture at $.70 Logic: The cost of the cashews plus the cost of the Brazil nuts should equal the total cost of the miture. Variables: Setup : Let B be the correct pounds of Brazil nuts and M be the correct pounds for the miture. Pounds of cashews + Pounds of Brazil nuts = Pounds of nut miture 0 pounds + B pounds = M pounds Setup : Cost of cashews + Cost of Brazil nuts = Cost of the nut miture $6.7 (0 lbs.) + $. (B lbs.) = $.70 (M lbs.) Eample: A boater travels 6 miles downstream in hours. Returning, against the current, takes 8 hours. What is the speed of the boat in still water and the speed of the river? Travels hours with current Travels 8 hours against current Distance of 6 miles Logic: Distance traveled = Rate of speed times length of Time elapsed. (D = R T) Downstream travel will add to your boat speed while upstream will subtract from your boat speed. Variables: Setup : Let B be the correct speed of the boat in still water and C the correct speed of the current. Distance downstream = Rate downstream Time downstream 6 miles = (B + C) hours Setup : Distance upstream = Rate upstream Time upstream 6 miles = (B C) 8 hours Valencia College All Rights Reserved 0 0 of 0 // : AM