MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006

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1 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 to place out of Math 092 or to prepare for the final examination in this course, one should be extremely comfortable with all of these items. NOTE: Calculators are not permitted in this course. The content listed below is covered in chapters through 7 in the second half of Pre and Introductory Algebra, A Combined Text by Aufman, Barker and Lockwood published by Houghton Mifflin Company), the current textbook used in Math 092. Adding, subtracting, multiplying, dividing, and exponentiating numbers integers, fractions and decimals); reducing fractions to lowest terms simplest form); converting between fractions, decimals, and percents. Evaluating and identifying variable terms, constant terms, and numerical coefficients of polynomials; determining what type of polynomial monomial, binomial, trinomial) a given expression is. Simplifying, adding, subtracting, multiplying, dividing, and exponentiating polynomials and rational expressions; rewriting zero and negative integer exponents. Simplifying arithmetic and algebraic expressions by following the order of operations Factoring by taking out the greatest common factor and by grouping; factoring trinomials, perfect square trinomials and differences of two squares. Solving algebraic equations involving integers, arithmetic and algebraic fractions, decimals, percents and/or parentheses; solving quadratic equations by factoring; solving ratios and proportions; solving literal equations for a specified variable in terms of other variables. Translating verbal expressions or sentences into variable expressions or algebraic equations. Solving application problems including, but not limited to, markup and markdown, determine-the-number problems, uniform motion, perimeters and areas, and ratio/rate problems). Graphing points and lines and identifying coordinates of given points on a rectangular Cartesian) coordinate system. Determining the x-intercepts, y-intercepts and slopes of straight lines. Finding equations of lines by using either the slope-intercept or the point-slope formula. This document was prepared by Ron Bannon using L A TEX 2ε and is a slight modification of Susan Gaulden s MTH 092 Content worksheet.

2 Review Problem Set At Essex County College you should be prepared to show all work clearly and in order, ending your work by boxing the answer. Furthermore, justify your answers algebraically whenever possible. These questions are for review only, and placement tests are not limited to these problems alone. Solutions and work are provided for each question. Please feel free to with questions or comments pertaining to this document. 2. Simplify each of the following. a) 3 + 2) + 6 4) 3 + 2) + 6 4) 3 + 2) b) ) 3 2 4) ) 3 2 4) ) c) ) ) 8.39) ) ) d) ) ) These are sample problems and you should not limit your study to just these problems. 2

3 e) f) g) ) ) ) ) Simplify each of the following and reduce to lowest terms simplest form). a) The LCD is x

4 b) Convert 7 8 into an equivalent decimal and percent. You may have to use long division % 4. Convert 5.2% into an equivalent fraction and decimal. 5.2% Evaluate b 2a) 2 + bc, given a 2, b 3, and c 4. b 2a) 2 + bc 3 2 2) ) 3 4) ) ) ) + 3 4) + 2) 4

5 6. Identify the constant terms) and the variable terms) of the expression 2a 2 8ab+b+0. Then identify the numerical coefficient of each variable term. The constant term is 0, the variable terms are 2a 2, 8ab and b. The numerical coefficient is 2a 2 is 2, the numerical coefficient is 8ab is 8, and the numerical coefficient is b is. 7. Simplify each of the following. 0a 3b 7a + 6b 5 a) 0a 3b 7a + 6b 5 0a 3b 7a + 6b 5 0a 7a 3b + 6b 5 7a + 3b 5 b) 2 x 2 3x ) + 3 4x 2 + ) 2 x 2 3x ) + 3 4x 2 + ) 2x 2 6x + 2x x 2 6x + 3 c) 5x 2 x + 8) 5x 2 x + 8) 5x 2x 6 3x 6 8. Simplify. a) 5m 2 2m + 2 ) 3 + 3m 2m 2) 5m 2 2m + 2 ) 3 + 3m 2m 2) 5m 2 2m m + 2m 2 7m 2 5m + 5 b) 2c 2 d 3) 3cd) 5c 3 d 5) 2c 2 d 3) 3cd) 5c 3 d 5) 2) 3) 5) c 2 c c 3) d 3 dd 5) 30c 6 d 9 5

6 c) 2n 2 ) 3 2n 2 ) 3 2n 2) 2n 2) 2n 2) 2) 2) 2) n 2) n 2) n 2) 8n 6 d) t 2 3t + 4 ) t 3) t 2 3t + 4 ) t 3) t 3) t 2 3t + 4 ) t t 2 3t + 4 ) 3 t 2 3t + 4 ) t 3 3t 2 + 4t 3t 2 + 9t 2 t 3 6t 2 + 3t 2 e) 3x ) 2x + 5) 3x ) 2x + 5) 6x 2 + 5x 2x 5 6x 2 + 3x 5 f) y 3) 2 y 3) 2 y 3) y 3) y 2 3y 3y + 9 y 2 6y + 9 g) 22m 2 n mn 33mn 2 mn 22m 2 n mn 33mn 2 mn 22m2 n mn mn mn 33mn2 mn 2m 3n h) x 2 5x + 8 ) x 3) Using the long division algorithm, please review this process if you cannot get this answer. x x 3. 6

7 i) 2y y 3 Again, using the long division algorithm, but you might find it helpful to rewrite 2y as 2y 2 + 0y + 7 first. 2y y Simplify. a) 2x + 9 9x x 5 5x The LCD is 45x. 2x + 9 9x x 5 5x 2x x 5 x 5 5x 9 9 0x x 45 45x 45x 0x x x x x b) x + + x x 6 5x 2 x 2 5x 6 Factor the denominators and the LCD is clearly x + ) x 6). x + + x x 6 5x 2 x 2 5x 6 x + ) + x x 6) 5x 2 x + ) x 6) x 6) x + ) x 6) + x x + ) x 6) x + ) 5x 2 x + ) x 6) x 6) + x x + ) 5x 2) x + ) x 6) x 6 + x2 + x 5x + 2 x + ) x 6) x 2 3x 4 x + ) x 6) x + ) x 4) x + ) x 6) x 4 x 6 7

8 c) x 2 + 3x 0 x 2 + 2x 8 x 2 + 3x 0 x 2 + 2x 8 x + 5) x 2) x + 4) x 2) x + 5 x + 4 d) x 2 7x 30 x 2 6x 40 2x2 + 5x + 2 2x 2 + 7x + 3 x 2 7x 30 x 2 6x 40 2x2 + 5x + 2 2x 2 + 7x + 3 x 0) x + 3) 2x + ) x + 2) x 0) x + 4) 2x + ) x + 3) x + 3) x + 2) x + 4) x + 3) x + 2 x + 4 e) 3x 2 y 9xy a 2 b 3x2 x 3 ab 2 3x 2 y 9xy a 2 b 3x2 x 3 ab 2 3x2 y 9xy a 2 b 3xy 3 x) a 2 b 3xy a 2 b 3by ax ab2 x 2 ab 2 3x 2 x 3 ab 2 x 2 3 x) 0. Simplify, final answer should not have negative expoments. a) 8x 5 y 2x 3 y 4 8x 5 y 2x 3 y 4 3x2 2y 3 8

9 b) 3ab 2 ) 2a b ) 3 3ab 2 ) 2a b ) 3 ) 3a 2b b 2 a ) 3a a ) 3 b 2 2b ) ) 3a a 3 b 2 8b 3 ) 3 3a4 8b 5 c) 2a 3 b 2 3ab 5 2a 3 b 2 3ab 5 2b2 b 5 3a 3 a 2b7 3a 4. Determine whether 9x 2 5 and y 2 + 3y 2 y of these. are monomials, binomials, trinomials, or none 9x 2 5 is a binomial, whereas y 2 + 3y 2 y is none of these.3 2. Factor each of the following polynomials as completely as possible. a) 2a 2 b 5a 2 b 2 + 7ab 2 2a 2 b 5a 2 b 2 + 7ab 2 ab 2a 5ab + 7b) b) 2x 2 5x 6xy + 5y 2x 2 5x 6xy + 5y x 2x 5) 3y 6x 5y) 2x 5) x 3y) 3 It s not a polynomial because there s a variable used as a divisor. 9

10 c) x 2 + 5x 36 x 2 + 5x 36 x + 9) x 4) d) 9x 2 3x 4 Not factorable over the integers. e) 6 9m 2 6 9m 2 4 3m) 4 + 3m) f) 4x 4 38x x 2 4x 4 38x x 2 2x 2 2x 2 9x + 24 ) 2x 2 2x 3) x 8) 3. Solve each of the following equations. a) 3 2x 5 + 4x 3 2x 5 + 4x 3 5 2x + 4x 2 6x 2 x So the solution is x 2. b) x Multiple both sides by the LCD. x x x x 2 So the solution is x /2. 0

11 c) 3 5 x x 2 x x 20 d) So the solution is x 20. x x x x e) So the solution is x x 4 So the solution is x 3/ x 4 6 8x x 3 8x 3 8 x f) 9x 3 2x ) 5 9x 3 2x ) 5 9x 6x x x 5 3 3x 2 x 4 So the solution is x 4.

12 g) x 2 8x x 2 8x x 5) x 3) 0 So the solution is x 5, or x 3. h) x x ) 2 x x ) 2 x 2 x 2 x 2 x 2 0 x 2) x + ) 0 So the solution is x 2, or x. i) x + 3) x + 4) 72 x + 3) x + 4) 72 x 2 + 7x x 2 + 7x 60 0 x + 2) x 5) 0 So the solution is x 2, or x 5. j) 5 x x So the solution is x x + 3 x 5 x ) 3 x + 3) 5x 5 3x + 9 5x 3x x 4 x 7 2

13 k) x 3 x x x x 3 x 3 2 x 3) x 3 2x 6 x 3 2x x 6 3 x 3 x x 3 x x 3 3 x 3 This equation has no solution. This may not be obvious, but the original statement is a contradiction and can never be true. 4. Solve for the indicated variable. a) R C S t for S R C S t Rt C S Rt C S C Rt S So the solution is S C Rt. b) A P + P rt for P So the solution is P A + rt. A P + P rt A P + rt) A + rt P 3

14 5. What is 35% of 80? 35% of What percent of 20 is 6? x x 6 5 x So the answer is 80% is.5% of what? x x 200 x 8. If 4 5x, evaluate x 2 3x +. First solve for x x 5 5x x Now evaluate x 2 3x + when x Is 5, 3) a solution of 3x 2y 9. Be sure to show enough work to justify your answer. No, because 3 5)

15 20. Determine the x and y-intercepts and the slope of the line given by 2x 3y 2. The x-intercept is 6, 0). The y-intercept is 0, 4). Using the formula for slope m y 2 y 4 0 x 2 x Determine the equation of the line that satisfies the given conditions. a) Contains the point 2, 3) and with slope m 2. y + 3 x 2) 2 2y + 6 x x + 2y Acceptable answers are: point-slope form y + 3 x 2) ; standard form x + 2y 4 ; 2 or slope-intercept form y 2 x 2. b) Contains the points 0, 6) and 2, 0). First determine the slope. m Acceptable answers are: point-slope form y 6 3 x 0) or y 0 3 x 2) ; standard form 3x + y 6 ; or slope-intercept form y 3x Translate each of the following phrases into variable expressions and simplify whenever possible. a) the product of t and the sum of t and 6 The translation is t t + 6), and the simplified form is t 2 + 6t. b) r decreased by the quotient of r and 3 The translation is r r 3, and the simplified form is 2r 3. c) nine less than the total of a number and two The translation is n + 2) 9, and the simplified form is n 7. 5

16 d) twice the difference of six times a number and seven The translation is 2 6n 7), and the simplified form is 2n Translate each of the following sentences into algebraic equations and solve to determine the numbers). Algebra must be used. a) The difference between five and twice a number is one. The translation is 5 2n, and the solution is n 2. b) Four less than three times a number is 25 The translation is 3n 4 25, and the solution is n 7. c) The sum of two numbers is eighteen. The total of three times the smaller and twice the larger is forty-four. The translation, where the small number is s and the larger number is 8 s, is 3s s) 44, and the solution is s 8, so the two numbers are 8 and 0. d) Find two consecutive even integers such that four times the first is three times the second. The translation, where the two consecutive even integers are n and n + 2, is 4n 3 n + 2), and the solution is n 6, so the two consecutive even intergers are 6 and 8. e) The sum of two numbers is six. twenty. The sum of the squares of the two numbers is The translation, where the two numbers are n and 6 n, is n n) 2 20, and the solution is n 4 or n 2, so the two consecutive even intergers are 2 and The high temperatures for a 6-day period in Barrow, Alaska, were 23 F, 29 F, 2 F, 28 F, 28 F, and 27 F. Calculate the average daily high temperature. Here the average is 23 F) + 29 F) + 2 F) + 28 F) + 28 F) + 27 F) 6 56 F 6 26 F. 25. The total cost to paint the inside of a house was $346. This cost included $25 for materials and $33 per hour for labor. How many hours of labor were required to paint the inside of the house? Algebra must be used. The translation, where h is the number of hours of labor, is h, and the solution is h 37, so 37 hours of labor were required. 6

17 26. A 2-foot board is cut into two pieces. Twice the length of the shorter piece is 3 feet less than the length of the longer piece. Find the length of each piece. Algebra must be used. The translation, where the short piece is s feet and the long piece is 2 s, is 2s 2 s) 3, and the solution is s 3, so the lengths are 3 feet and 9 feet. 27. The perimeter of a rectangle is 26 meters. The length of the rectangle is meter more than twice the width. Find the length and width of the rectangle. Algebra must be used. The translation, where the width is w meters and the length piece is 2w +, is w + ) + 2w, and the solution is w 4, so the length and width are are 9 meters and 4 meters. 28. The perimeter of a triangle is 0 inches. One side is twice the second side. The third side is 30 inches more than the second side. Find the length of each side. Algebra must be used. The translation, where the length of the second side is x inches and the other sides 2x inches and x + 30 inches, is x + 2x + x + 30) 0, and the solution is x 20, so the lengths are 20 inches, 40 inches, and 50 inches. 29. The width of a rectangle is 5 feet less than the length. The area of the rectangle is 76 square feet. Find the length and width of the rectangle. Algebra must be used. The translation, where the length is l feet and the width is l 5 feet, is l l 5) 76, and the solution is l 6 or l 4, so the length and width are 6 feet and feet. 30. On a map, two cities are inches apart. If 3 8 inch on the map represents 25 miles, determine the number of miles between the two cities. First convert inches to an improper fraction mi in 3/8 in mi 7 25 mi 75 miles A car leaves a town traveling at 40 mph. Two hours later, a second car leaves the same town, on the same road, traveling at 60 mph. In how many hours will the second car pass the first car? Algebra must be used. The translation, where t represents the time, in hours, the second car has traveled when the two meet, is 60t 40 t + 2), and the solution is t 4, so it will take 4 hours. 4 The is an extraneous solution. 7

18 32. Two cars, one traveling 0 mph faster than the other, start at the same time from the same point and travel in opposite directions. In 3 hours they are 300 miles apart. Determine the rate of each car. Algebra must be used. The translation, where the rate of the slower car is r and the rate of the faster car is r + 0, in mies per hour, is 3r + 3 r + 0) 300, and the solution is r 45, rates are 45 mph and 55 mph. 33. The sales tax on a car that sold for $2,000 is $780. At this rate, how much higher is the sales tax on a car that sells for $3,500? The translation, where the length is x represents the additional tax paid, is x 3500, and the solution is x 97.50, so it will take $ more. 34. Graph the following lines on a Cartesian rectangular) coordinate system. a) y 2 3 x + Two points, such as 0, ) 3, ) should be used to plot the graph Figure : y 2 3 x + 8

19 b) x 2y 4 Two points, such as 0, 2) 4, 0) should be used to plot the graph Figure 2: x 2y 4 c) x 2 Two points, such as 2, 0) 2, ) should be used to plot the graph Figure 3: x 2 9

20 d) y 3 0 Two points, such as 0, 3), 3) should be used to plot the graph Figure 4: y