Solution set consists of one or more solutions (but not an identity). x = 2 or x = 3
|
|
- Peregrine Foster
- 7 years ago
- Views:
Transcription
1 . Linear and Rational Equations In this section you will learn to: recognize equations that are identities, conditional, or contradictions solve linear equations in one variable find restrictions on variable values solve rational equations with variables in the denominator solve formulas for a specific value Types of Equations Identity Conditional Contradiction/Inconsistent Solution set consists of all real numbers*. Eample: + = 5 + = (or 0 = 0) (Both sides are identical.) Solution set consists of one or more solutions (but not an identity). Eample: = 0 = or = The equation has no solutions. Eample: + = 6 + = (Since, there is no solution.) ALWAYS TRUE SOMETIMES TRUE NEVER TRUE *ecept when one or both sides of the equation results in division by 0 (Eample: = ) Solving Linear Equations ( a + b = 0, where a and b are real numbers and a 0 ) Eample : Solve ( 6) = 6 Steps: (if coefficients are integers). Simplify each side of the equation.. Move variable terms to one side and constant terms to the other.. Isolate the variable. (Multiply or divide based on the coefficient.) 4. Check the solution in the original equation. Page (Section.)
2 Eample : Solve 5 ( + ) = + ( 5) Eample : Solve: + = 5( + ) + 6 (NOTE: a b is equivalent to b a.) Steps: (if coefficients are fractions). Find the LCD (Lowest Common Denominator). Apply distributive property first if necessary.. Multiply both sides by the LCD. 6. Apply Steps -4 in Eample. Finding Variable Restrictions Eample 4: Find the restrictions on the variable a in each of the equations below. (a) a = a 5 a + 5 (b) a = a 4 a 5 a Solving Rational Equations (equations having one or more rational epressions) 5 0 Eample 5: Solve = + 4 Steps: (if variable is in denominator). Eclude any values that cause a zero denominator (restrictions on the variable). Page (Section.). Find the LCD and multiply both sides by the LCD.. Simplify and solve the equation. 4. Check your solution in the original equation.
3 Page (Section.) Eample 6: Find all values of for which y y = given 6 = y and = y. Eample 7: Solve for n: = + n n n n Eample 8 (optional): Solve: + = + +
4 Solving Formulas for a Specific Value Steps: Eample 9: Solve V = Bh for B.. If fractions are involved, multiply by LCD.. Move terms with desired variable to one side. Move all other terms to other side.. If two or more terms contain the desired variable, FACTOR out the variable. 4. Divide both sides by the non variable factor. Eample 0: Solve S = P + Prt for t. Eample : Solve S = P + Prt for P. Eample : Solve a b = for b. Eample : Solve c w = y + for z. z Eample 4: Solve A + B C = for A. Eample 5: Solve + = 5 for. A B y Page 4 (Section.)
5 . Homework Problems. Find restrictions on the variable in each equation below: (a) = 5 4 (b) ( + 4) = + 4 (c) + = 7 For Problems 8, solve the equations and classify each as an identity, contradiction or conditional.. 4(5 + ) = = = = ( + ) = = 7( ) = + 9. SolveC = π r for r. 0. Solve a = b c for c. a y. Solve = b for.. Solve A + B C = for B. A B. Solve AC + D A + B = for D. 4. Solve: D = for b. a b c 5 5. Solve C = ( F ) for F. 9. Homework Answers:.(a) 0, ± (b) 4, 0, (c) 0, ±. {-}; conditional {}; conditional 4. ; conditional 5. φ ; contradiction ; conditional C b a ab 7. all real numbers; identity 8. {}; conditional 9. r = 0. c =. = π b + y AC A AC ac 9. B =. D = 4. b = 5. F = C + C + A + B c a 5 Page 5 (Section.)
6 . Applications of Linear Equations In this section you will learn to: use linear equations to solve word problems Steps/Tips for Solving Word Problems:. Read the problem carefully. Underline key words and phrases. Let (or any variable) represent one of the unknown quantities. Draw a picture or diagram if possible.. If necessary write epressions for other unknowns in terms of.. Write a verbal model of the problem and then replace words with numbers, variables and/or symbols. 4. Solve the equation. Answer the question in the problem. Label answers! 5. Check your answer(s) in the original word problem (not in your equation). Eample : The number of cats in the U. S. eceeds the number of dogs by 7.5 million. The number of cats and dogs combined is 4.7 million. Determine the number of dogs and cats in the U. S. Eample : After a 0% discount, a cell phone sells for $. Find the original price of the cell phone before the discount was applied to the purchase. Eample : Including a 6% sales ta, an item costs $9.69. Find the cost of the item before the sales ta was added. Eample 4: You are choosing between two car rental agencies. Avis charge $40/day plus $.0/mile to rent a car. Hertz charges $50/day plus $.08/mile. You plan to rent the car for three days. After how many miles of driving will the total cost for each agency be the same? Page (Section.)
7 Eample 5: The perimeter of a triangular lawn is 6 meters. The length of the first side is twice the length of the second side. The length of the third side is 6 meters shorter than three times the length of the second side. Find the dimensions of the triangle. Eample 6: (Simple Interest Problem) Tricia received an inheritance of $5500. She invested part of it at 8% simple interest and the remainder at % simple interest. At the end of the year she had earned $540. How much did Tricia invest at each interest amount? Eample 7: How many liters of a 9% solution of salt should be added to a 6% solution in order to obtain 50 liters of a % solution? Eample 8: A student scores 8%, 86% and 78% on her first three eams. What score is needed on the fourth eam for the student to have an average of 85% for all four eams? Eample 9: A student scores 85%, 7%, 96%, and 98% for his first four chapter eams. If the fifth eam, the final eam, counts twice as much as each of the chapter eams, is it possible for the student to get a high enough final eam score to get a 90% average for the course? Page (Section.)
8 Eample 0: A rectangular swimming pool measures 8 feet by 0 feet and is surrounded by a path of uniform width around all four sides. The perimeter of the rectangle formed by the pool and the surrounding path is feet. Determine the width of the path. Eample : Sam can plow a parking lot in 45 minutes. Eric can plow the same parking lot in 0 minutes. If Sam and Eric work together, how long will it take them to clear the lot? Eample : A plane leaving Lansing International Airport travels due east at a rate of 500 mph. A second plane takes off 5 minutes later traveling in the same direction at 650 mph. How long will it take for the second plane to catch up to the first plane? Page (Section.)
9 . Homework Problems. When a number is decreased by 0% of itself, the result is 56. What is the number?. 5 % of what number is.6?. 5 is what % of 80? 4. Find.5% of $ After a 0% price reduction, a cell phone sold for $77. Find the original price of the phone. 6. Find two consecutive even integers such that the sum of twice the smaller integer plus the larger is The length of a rectangular garden plot is 6 feet less than triple the width. If the perimeter of the field is 40 feet, what are its dimensions? 8. Trinity College currently has an enrollment of,00 students with a projected enrollment increase of 000 students a year. Brown college now has 6,800 students with a projected enrollment decline of 500 students per year. Based on these projections, when will the colleges have the same enrollment? 9. Sam invested $6,000 in two different stocks. The first stock showed a gain of % annual interest while the second stock suffered a 5% loss. If the total annual income from both investments was $40, how much was invested at each rate? 0. How many liters of a 7% acid solution should be added to 0 liters of a 5% solution in order to obtain a 0% solution?. How many liters of skim milk (0% fat) must be added to liters of milk containing.5% butterfat in order to dilute the milk to % butterfat?. Amy scored 78%, 64%, 98%, and 88% on her first four eams. What score does she need on her fifth eam in order to have an 85% average for all five eams?. Scott showed improvement on his five math eams throughout the semester improving by % on each successive eam. If the fifth eam (final eam) counted twice as much as the first four eams and his average was 79% for all five eams, what score did he receive on his first eam? 4. The length of a rectangular tennis court is 6 feet longer than twice the width. If the perimeter of the court is 8 feet, find the dimensions of the court. 5. During a road trip, Tony drove one-third the distance that Lana drove. Mark drove 4 more miles than Lana. The total distance they drove on the trip was 46 miles. How many miles did each person drive? 6. A garden hose can fill a swimming pool in 5 days. A larger hose can fill the pool in days. How long will it take to fill the pool using both hoses? 7. The Smith family drove to their vacation home in Michigan in 5 hours. The trip home took only hours since they averaged 6 mph more due to light traffic. How fast did they drive each way?. Homework Answers: % $ and ft by 6 ft 8. 9 years 9. %; 5% liters..5 liters. 97%. 7% 4. 6 ft by 78 ft 5. Tony: 46 miles; Lana: 8 miles; Mark:6 miles days (45 hrs) 7. 9 mph; 65 mph Page 4 (Section.)
10 . Quadratic Equations In this section you will learn to: solve quadratics equations by. factoring. square root property. quadratic formula 4. completing the square 5. graphing (used mainly for checking not considered an algebraic solution) use the discriminant to find the number and type of solutions (roots, -intercepts, zeros) A quadratic equation in is an equation that can be written in the general form a + b + c = 0, where a, b, and c are real numbers, with a 0. A quadratic equation in is also called a second-degree polynomial equation. The Zero-Product Principle: If the product of two algebraic epressions is zero, then at least one of the factors equal to zero. If AB = 0, then A = 0 or B = 0. Solving Quadratic Equations by Factoring: Eample : 9 = Eample : = + 0 Steps:. Rearrange equation so that one side is 0. (Look for GCF.). Factor. (Use sum/product idea when a =. If a, use grouping*.). Set each factor equal to Solve each equation. 5. Check in original equation or by graphing (observe -intercepts). *Go to Class Pages on math home page ( for steps using Grouping Method. Page (Section.)
11 Eample : 4 = (Use guess & check or grouping method.) Solving Quadratic Equations by the Square Root Method: Square Root Property: If a > 0 then = a has two real roots: = a or = a Reminder: Any time you choose to take a square root when solving an equation, you must include ±. (Eample: If = 4, then = ±.) Eample 4: = 47 Eample 5: (8 ) = 5 Solving Quadratic Equations Using the Quadratic Formula: Quadratic Formula: If a + b + c = 0, where a 0, then b ± b 4ac a =. Eample 6: Solve and simplify: = 5 Page (Section.)
12 Eample 7: Solve and simplify: = Eample 8 (optional): Solve for t and simplify: h = t 6t In the quadratic formula, Beware: The discriminant is NOT b ± b 4ac =, the value of b 4ac is called the discriminant. a b 4ac!!! b 4ac > 0 b 4ac < 0 b 4ac = 0 Eample 9: Determine the number and type of solutions for the equations below. (Do not solve.) (a) = (b) = 70 Page (Section.)
13 Solving a Quadratic Equation by Completing the Square: Recall: Perfect Square Trinomials ( + ) = = ( + 6) 8 + = ( ) = ( ) + 00 = ( ) Eample 0: + 5 = 0 Steps:. Divide each term by the coefficient of.. Move the constant term to the right side.. Form a perfect square trinomial by adding (/ of the new coefficient of ) to both sides. 4. Factor the left side (perfect square trinomial). Add the terms on the right side. 5. Finish solving using the Square Root Method. Page 4 (Section.)
14 . Homework Problems Solve Problems -6 by factoring:. = 5. 5 =. 0 = a ( a ) 5 = = 0 6. m = 7m + 6 Solve Problems 7-9 by using the quadratic formula: = = = + 7 Solve Problems 0- using the square root method: 0. 5 =. ( + ) + 5 =. ( 4) = 8 49 y. Solve for t: h = 6t 4 4. Solve for : = b a For Problems 5-7, determine the number and type of solutions by eamining the discriminant. 5. = = = 4 Solve each of the following quadratic equations by completing the square: = 0 9. = = Homework Answers:. 0. {-, 5}. {-5} 4. {-, 5} 5. {-, 5} 6., ± ± ± h {, 5} ±. φ.. ± a b y ± 5. 0 real roots 6. real roots 7. real root 8. {-5, -} 9. {-, 5} b 0. ± Page 5 (Section.)
15 .4 Application of Quadratic Equations In this section you will learn to: solve rational equations by changing to quadratic form use quadratic equations to solve word problems Solving Rational Equations: Steps: Eample : 4 =. Multiply both sides by the LCD.. Simplify both sides.. Change to quadratic form: a + b + c = Solve using a quadratic technique. 5. Check answer in original equation. (Be aware of restrictions!) Eample : + = Eample : The length of a rectangle eceeds its width by feet. If its area is 54 square feet, find its dimensions. Page (Section.4)
16 Eample 4: The MSU football stadium currently has the 4 th largest HD video screen of any college stadium. The rectangular screen s length is 7 feet more than its height. If the video screen has an area of 5760 square feet, find the dimensions of the screen. (MSU math fact: The area of the video screen is about 600 ft larger than Breslin s basketball floor.) Eample 5: When the sum of 8 and twice a positive number is subtracted from the square of the number, the result is 0. Find the number. Eample 6: Find at least two quadratic equations whose solution set is, 5. Eample 7: The height of an object thrown upward from the roof of a building 00 feet tall, with an initial velocity of 00 feet/second, is given by the equation h = 6t + 00t + 00, where h represents the height of the object after t seconds. How long will it take the object to hit the ground? (Round answer to nearest hundredth.) Page (Section.4)
17 Eample 8: In a round-robin tournament, each team is paired with every team once. The formula below models the number of games, N, that must be played in a tournament with teams. If 55 games were played in a round-robin tournament, how many teams were entered? N = Eample 9: John drove his moped from Lansing to Detroit, a distance of 0 km. He drove 0 km per hour faster on the return trip, cutting one hour off of his time. How fast did he drive each way? Eample 0 (optional): When tickets for a rock concert cost $5, the average attendance was 00 people. Projections showed that for each 50 decrease in ticket prices, 40 more people would attend. How many attended the concert if the total revenue was $7,80? Page (Section.4)
18 .4 Homework Problems. Solve: = +. Solve: = +. The base of a triangle eceeds its height by 7 inches. If its area is 55 square inches, find the base and height of the triangle. 4. A regulation tennis court for a doubles match is laid out so that its length is 6 feet more than two times its width. The area of the doubles court is 808 square feet. Find the length and width of a doubles court. 5. If 0 games were played in a round-robin tournament, how many teams were entered? (Refer to Eample 9 in class notes for formula.) 6. A quadratic equation has two roots: ¾ and -5. (a) Find a quadratic equation where the coefficient of the term is. (b) Find a second equation that has only integers as coefficients. 7. The height of a toy rocket launched from the ground with an initial velocity of 8 feet/second, is given by the equation h = 6t + 8t, where h represents the height of the rocket after t seconds. How long will it take the rocket to hit the ground? (Round answer to nearest hundredth.) 8. The height of an object thrown upward from the roof of a building 00 feet tall, with an initial velocity of 00 feet/second, is given by the equation h = 6t + 00t + 00, where h represents the height of the object after t seconds. At what time(s) will the object be 00 feet above the ground? (Round answer to nearest hundredth.) 9. Jack drove 600 miles to a convention in Washington D. C. On the return trip he was able to increase his speed by 0 mph and save hours of driving time. (a) Find his rate for each direction. (b) Find his time for each direction. 0. When tickets for a rock concert cost $, the average attendance was 500 people. Projections showed that for each $ increase in ticket prices, 50 less people would attend. At what ticket price would the receipts be $ Homework Answers:. { 0} ±. {0, }. base: in; height: 5 in ft by 6 ft teams 6. (a) + = 0 ; (b) = 0 (answers vary) 7. 8 seconds and 5 seconds 9. (a) 40 mph; 50 mph (b) 5 hours; hours 0. $4 ($ increase) or $8 ($4 decrease) Page 4 (Section.4)
19 .5 Comple Numbers In this section you will learn to: add, subtract, multiply and divide comple numbers simplify comple numbers find powers of i solve quadratic equations with comple roots The imaginary unit i is defined as i =, where i =. The set of all numbers in the form a + bi with real numbers a and b and i, the imaginary unit, is called the set of comple numbers. (Standard form for a comple number is a + bi.) Rational Numbers THE COMPLEX NUMBER SYSTEM Real Numbers Irrational Numbers Imaginary Numbers Eample : Simplify each imaginary number below: 9 = 6 = 8 = 7 = = 5 Equality of Comple Numbers: a + bi = c + di if and only if a = c and b = d. Addition of Comple Numbers: ( a + bi) + ( c + di) = ( a + c) + ( b + d) i Subtraction of Comple Numbers: ( a + bi) ( c + di) = ( a c) + ( b d) i Multiplication of Comple Numbers: ( a + bi)( c + di) = ( ac bd) + ( ad + bc) i NOTE: Add, subtract, or multiply comple numbers as if they were binomials. Page (Section.5)
20 Eample : Add: ( i) + (-4 + 5i) Eample : Subtract: ( i) (-4 + 5i) Eample 4: Multiply: ( i)(-4 + 5i) Recall: (a + b) and (a b) are conjugates. Therefore, (a + b)(a b) = a b. a + bi and a bi are called comple conjugates therefore: ( a + bi)( a bi) = Eample 5 (Division): Simplify i 4 + 5i (Hint: Multiply numerator and denominator by conjugate of the denominator.) Eample 6 (Division): Find the reciprocal of 4 + i. Page (Section.5)
21 Eample 7: Simplify each of the following powers of i. i = 4 i = i = 0 i = i = 6 i = Recall: = i. Therefore b = ( ) b = b = i b. Eample 8: Simplify and write each answer in standard form. (a) (b) 4 9 (c) ( + 5) (d) ( + )( ) (e) + Eample 8: Solve: a + 4a + 8 = 0 Eample 9: Solve: 5 + = 5 Page (Section.5)
22 .5 Homework Problems. Simplify each epression: (a) 69 (b) 00 (c) 08 (d) (e) 7 6 (f) 4 5 i (g) 6 i (h) 7 i Perform the indicated operation for each problem below. Simplify each problem and write the answer in standard form. (a) ( 7i) + ( 8 4i) (b) ( 6 i ) ( 9 + 5i) (c) ( 4i )( 8 + 5i) (d) ( 4 i 5 ) (e) 5 i (f) i + i (g) i + i (h) ( + i) (i) (j) 6 5 (k) 7 (l) ( 5i) (m) ( 4) (n) Simplify: 5 ( + i) 4. Simplify: ( + i)( i) + i 5. Find the reciprocal of 7 + i. 6. Solve each of the following: (a) = 7 (b) + 5 = 6 (c) = 0 (d) + = Homework Answers:. (a) i (b) 0i (c) 6i (d) i (e) 9i (f) i (g) - 7 (h) i. (a) -5 i (b) 5 7i (c) i (d) -9 40i (e) -5i (f) i 4 (g) i (h) + i (i) i (j) -0 (k) -9 (l) + 0i (m) + i (n) i ± i i 6. (a) ± 4i ; (b) ± i ; (c) or ± ; (d) 5 ± i Page 4 (Section.5)
23 .6 Polynomial & Radical Equations In this section you will learn to: solve polynomial equations using factoring solve radical equations solve equations with rational eponents solve equations in quadratic form A polynomial equation of degree n is defined by a n n + a n n a + a + a0 = 0, where n is a nonnegative integer (whole number), an, an, an,... a, a, a0 are real numbers, a 0, n a is called the leading coefficient, n a 0 is called the constant term (Recall: 0 0 =, for 0, therefore a 0 = a0 ), The degree of the polynomial is n (the highest degree of any term in the polynomial). Solving Polynomials by Factoring: Eample : Solve: 4 4 = Eample : Solve: 9y + 8 = 4y + 8y Eample : Solve: 6 = 0 Page (Section.6)
24 Solving Radical Equations: Eample 4: Solve: + = Steps:. Isolate radical(s).. Square both sides.. Epand. 4. Solve for. 5. Check answer! Eample 5: Solve: + 5 = (Hint: Move one of the radicals to the other side and repeat Eample 4 Steps - after epanding.) Solving Equations in Quadratic Form (using substitution): Eample 6: Solve: 0 = 0 Eample 7: Solve: 7 8 = 0 Page (Section.6)
25 5 Eample 8: Solve: 5 4 = 0 Eample 9: Solve: 4 = 9 Eample 0: Try this alternative shortcut to substitution using this eample for the following problems: 0 = 0 ( 5)( + ) = 0 = 5 or = 4 0 = 0 0 = 0 0 = 0 0 = 0 ( + ) ( + ) 0 = 0 Page (Section.6)
26 Solving Equations with Rational Eponents: Recall: if 0 = = For eample: ( 5) = 5 = 5 if < 0 m n m =, =, =, = n 5 8 Eample : Solve: 4 = 0 Eample : Solve: ( + 5) = 4 Page 4 (Section.6)
27 .6 Homework Problems Solve each of the equations below using any appropriate method: = = = 4. 4 = = 6. + = = = = = 4. + = +. 4 = 50. = = 0 5. ( 7) = = = Solve each of the following equations using substitution: 8. = = 0. 8 = Homework Answers:. {, 5} 7. 9, 4 0 ±. {0,, }. {-, 4} 4. { } 8. {-6, } 9. φ 0. { ± 4}. {-}. {-5, 0, 5}. ± 5. {, -} 6. {},, 4. {6} 5. {-, 5} 6. {, ± } 7. 5, 8., , 4 0. {4} Page 5 (Section.6)
28 .7 Inequalities In this section you will learn to: use interval notation understand properties of inequality solve linear (and compound) inequalities solve polynomial inequalities solve rational inequalities Interval Notation: Inequality Graph Interval Notation Set Builder Notation < 4 < or > 5 all real # s Intersection Union Page (Section.7)
29 If Properties of Inequalities +/- Property of Inequality Multiplication/Division Property of Inequality a < b, then a + c < b + c a c < b c (Adding or subtracting does not affect the > or < sign.) c > 0 (c is positive) If a < b, then ac < bc a < c b c (Multiplying or dividing by a positive number does not affect the > or < sign.) c < 0 (c is negative) If a < b, then ac > bc a b > c c (When multiplying or dividing by a negative number, reverse the > or < sign.) Eample : Solve and graph the inequality below. Write the answer using interval notation. Eample : Solve and graph the inequality below. Write the answer using interval notation. 5 5 (7 + 9) > ( 5) Eample : Solve the compound ( and ) inequality < + 5. Write answer using interval notation. (a) Solve by isolating the variable. (b) Solve by writing each inequality separately. Eample 4: Avis charges $40/day plus $.0/mile to rent a car. Hertz charges $50/day plus $.08/mile. When is Avis a better deal if you are renting a car for three days? Page (Section.7)
30 A polynomial inequality is any inequality of the form: f ( ) < 0 (graph is below the -ais) f ( ) 0 (graph is on or below the -ais) f ( ) > 0 (graph is above the -ais) f ( ) 0 (graph is on or above the -ais) where f is a polynomial function. Eample 5: Solve 6 > 5. Write the solution using interval notation. Steps for Solving Polynomial Inequalities:. Epress as f() > 0 or f() < 0. (Get 0 on right side.). Set f() = 0 and solve for to get Boundary Points.. Plot the boundary points on a number line to obtain Intervals. 4. Test Values within each interval and evaluate f () for each value. If f () > 0, then f () is + for interval. If f () < 0, then f () is for interval. 5. Write the solution using interval notation. Check the solution on your calculator. Eample 6: Solve Write the solution using interval notation. Page (Section.7)
31 Eample 7: Solve Write the solution using interval notation. A rational inequality is any inequality of the form: f ( ) < 0 (graph is below the -ais) f ( ) 0 (graph is on or below the -ais) f ( ) > 0 (graph is above the -ais) f ( ) 0 (graph is on or above the -ais) p( ) where f is a rational function. ( f ( ) =, where p and q are polynomials and q ( ) 0) q( ) Eample 8: Solve 0. Write the solution using interval notation. + 5 Steps for Solving Rational Inequalities:. Epress as f () > 0 or f () < 0. (Get 0 on right side.) *. Find values that make the numerator & the denominator = 0. These are the Boundary Points. (Note Restrictions!). Plot the boundary points on a number line to obtain Intervals. 4. Test Value within each interval and evaluate f () for each value. If f () > 0, then f () is + for interval. If f () < 0, then f () is for interval. 5. Write the solution using interval notation. Check the solution on your calculator. Page 4 (Section.7)
32 Eample 9: Solve +. Write the solution using interval notation. Eample 0: A ball is thrown vertically from a rooftop 40 feet high with an initial velocity of 64 feet per second. During which time period will the ball s height eceed that of the rooftop? (Use h ( t) = 6t + v0t + s0 where v 0 = initial velocity, s 0 = initial height/position, and t = time. (You may also want to graph this function on your calculator using the viewing rectangle [0, 0, ] by [-00, 500, 00]). Page 5 (Section.7)
33 .7 Homework Problems Solve each of the inequalities below and write the answer using interval notation ( + ). ( + ) 4( + 5). ( + ) ( + 7) < 4. ( 5) 5. + > + ( + ) 6. + < < > < 0 4. < > , 4., 5 5., 7 7 ( 7. [-, 5) 8. (, ) (4, ) 9., 0. (, ] [, ]. {0} [ 9, ) 4 (, ]. φ 4. (, 0), 5. ( ) ( 4, ),,.7 Homework Answers:. [, ). [, ) 6., ). { }, 6. [ ) 7. [,4) 8. (, 5] (, ) 9., 0. 5, Page 6 (Section.7)
34 .8 Absolute Value In this section you will learn to: solve absolute value equations solve absolute value inequalities apply absolute inequalities The absolute value of a real number is denoted by, and is defined as follows: If 0 (non-negative number), then =. If < 0 (negative number), then = -. Absolute Value Equations: If k 0 and = k, then = k or = -k. Solving Absolute Value Equations: Eample : Solve: + = 7 Eample : Solve: = 5 Eample : Solve: = Eample 4: Solve: 6 = Eample 5: Solve: = 4 Page (Section.8)
35 Eample 6: Solve (using guess and check) and graph each of the following: = > < Which solutions involve and? Which use the concept of or? Absolute Value Inequalities: If k > 0, then k is equivalent to k or k (union). Solving Absolute Value Inequalities: If k > 0, then k is equivalent to k and k (intersection). Eample 7: Solve and graph the inequality below. Write answer in interval notation. Eample 8: Solve and graph the inequality below. Write answer in interval notation. 5 4 < 5 + Eample 9: Solve: + 8 Eample 0 (optional): Solve: 0 < + 5 Page (Section.8)
36 Eample : The temperatures on a summer day satisfy the inequality t 74 0, where t is the temperature in degrees Fahrenheit. Epress this range without using absolute value symbols. Eample : A weight attached to a spring hangs at rest a distance of inches off of the ground. If the weight is pulled down (stretched) a distance of L inches and released, the weight begins to bounce and its distance d off of the ground at any time satisfies the inequality d L. If equals 4 inches and the spring is stretched inches and released, solve the inequality to find the range of distances from the ground the weight will oscillate. Page (Section.8)
37 .8 Homework Problems: Solve each of the absolute value equations below: = = m 7 + = 5. 5 = = Solve each inequality. Write the solution using interval notation < > < 7. >. A Steinway piano should be placed in room where the relative humidity h is between 8% and 7%. Epress this range with an inequality containing an absolute value.. The optimal depths d (in feet) for catching a certain type of fish satisfy the inequality 8 d < 0. Find the range of depths that offer the best fishing..8 Homework Answers:. {, 9}. {-6, -}. {-.5, } 4. {, 5} 5. {-5, -, 5, 7} 6. (-5, -) 7. (, 9) 8. (, ], 9. [0, 0] 0. φ. (, 0) (, ). h 55 < 7. (00,400) Page 4 (Section.8)
Review of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationSummer Math Exercises. For students who are entering. Pre-Calculus
Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationSECTION 1-6 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More informationVeterans Upward Bound Algebra I Concepts - Honors
Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More information7 Literal Equations and
CHAPTER 7 Literal Equations and Inequalities Chapter Outline 7.1 LITERAL EQUATIONS 7.2 INEQUALITIES 7.3 INEQUALITIES USING MULTIPLICATION AND DIVISION 7.4 MULTI-STEP INEQUALITIES 113 7.1. Literal Equations
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationSect 6.7 - Solving Equations Using the Zero Product Rule
Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More information7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More information( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:
Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationWarm-Up Oct. 22. Daily Agenda:
Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment
More informationMTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationMATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationALGEBRA I FINAL EXAM
ALGEBRA I FINAL EXAM A passing score of 9 on this test allows a student to register for geometry. JUNE 00 YOU MAY WRITE ON THIS TEST . Solve: 7= 6 6 6. Solve: =. One tai cab charges $.00 plus 7 cents per
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationStudents will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationIOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.
IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of $2,000. She also earns $500 for
More informationMidterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013
Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationThe majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationEquations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationMTH 100 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created June 6, 2011
MTH 00 College Algebra Essex County College Division of Mathematics Sample Review Questions Created June 6, 0 Math 00, Introductory College Mathematics, covers the mathematical content listed below. In
More informationAnchorage School District/Alaska Sr. High Math Performance Standards Algebra
Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,
More informationALGEBRA I (Common Core) Tuesday, June 3, 2014 9:15 a.m. to 12:15 p.m., only
ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Tuesday, June 3, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The
More information10 7, 8. 2. 6x + 30x + 36 SOLUTION: 8-9 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial.
Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationAlgebra 1. Practice Workbook with Examples. McDougal Littell. Concepts and Skills
McDougal Littell Algebra 1 Concepts and Skills Larson Boswell Kanold Stiff Practice Workbook with Examples The Practice Workbook provides additional practice with worked-out examples for every lesson.
More informationChapter 4 -- Decimals
Chapter 4 -- Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationMATH 100 PRACTICE FINAL EXAM
MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
More informationSECTION 1.6 Other Types of Equations
BLITMC1B.111599_11-174 12//2 1:58 AM Page 11 Section 1.6 Other Types of Equations 11 12. A person throws a rock upward from the edge of an 8-foot cliff. The height, h, in feet, of the rock above the water
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationFlorida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District
Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More information12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2
DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials
More informationFSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers
FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward
More informationVirginia Placement Test Practice Questions and Answers
Virginia Placement Test Practice Questions and Answers Table of Contents Practice Problems for UNIT 1 Operations with Positive Fractions... 1 Practice Problems for UNIT Operations with Positive Decimals
More information6706_PM10SB_C4_CO_pp192-193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
More informationAlgebra Word Problems
WORKPLACE LINK: Nancy works at a clothing store. A customer wants to know the original price of a pair of slacks that are now on sale for 40% off. The sale price is $6.50. Nancy knows that 40% of the original
More information1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.
Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1-.
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationSolving Rational Equations and Inequalities
8-5 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More informationLesson 1: Multiplying and Factoring Polynomial Expressions
Lesson 1: Multiplying and Factoring Polynomial Expressions Student Outcomes Students use the distributive property to multiply a monomial by a polynomial and understand that factoring reverses the multiplication
More informationUse Square Roots to Solve Quadratic Equations
10.4 Use Square Roots to Solve Quadratic Equations Before You solved a quadratic equation by graphing. Now You will solve a quadratic equation by finding square roots. Why? So you can solve a problem about
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationgiven by the formula s 16t 2 v 0 t s 0. We use this formula in the next example. Because the time must be positive, we have t 2.64 seconds.
7 (9-0) Chapter 9 Quadratic Equations and Quadratic Functions where x is the number of years since 1980 and y is the amount of emission in thousands of metric tons (Energy Information Administration, www.eia.doe.gov).
More information7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More informationPERT Computerized Placement Test
PERT Computerized Placement Test REVIEW BOOKLET FOR MATHEMATICS Valencia College Orlando, Florida Prepared by Valencia College Math Department Revised April 0 of 0 // : AM Contents of this PERT Review
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationMath 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction
Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationAlgebra 1. Curriculum Map
Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More information