Review for College Mathematics A SELECTION OF PROBLEMS INTENDED TO MAINTAIN STUDENTS ESSENTIAL ALGEBRA SKILLS INSTRUCTOR COPY Equations and Modeling 1 A moving company charges $46 to move a certain machine 10 miles and $58 to move the same machine 30 miles. a) b) c) d) Find an equation that defines this relationship if it is linear. What will it cost to move the machine 25 miles? What is the minimum charge for moving the machine? What is the rate for each mile the machine is moved? Solution: a) C = 0.6 m + 14 b) $55 c) $40 d) $0.60 per mile MATHEMATICS for COLLEGE READINESS Equations and Modeling 2 The total cost of producing a certain item consists of paying rent for the building and paying a fixed amount per unit for material. The total cost is $250 it 10 units are produced and $330 if 30 units are produced. a) Find the equation that defines this relationship if it is linear. b) What will it cost to produce 100 units? c) How much is paid in rent? d) What is the cost of the material for each unit? Solution: a) C = 4x + 210, where x = number of units produced. b) $610 c) $210 d) $4 per unit MATHEMATICS for COLLEGE READINESS Equations and Modeling 3 If x2 + 2x = 3, demonstrate algebraically which of the following is a possible value for x. a) – 3 –2 c) –1 d) 0 e) 3 b) Solution: a) – 3 MATHEMATICS for COLLEGE READINESS Equations and Modeling 4 How many solutions are there to the systems of equations? Solve algebraically. x 2y 6 y x 1 3 6 Solution: Infinite: the lines coincide so the solution is x, y : x 2 y 6 MATHEMATICS for COLLEGE READINESS Equations and Modeling 5 The sum of the square of a number and the product of the number and 3 equals 40. a) Write an equation to model this situation. b) What are the positive answer(s)? Solution: a) b) n 3n 40 2 n =5 MATHEMATICS for COLLEGE READINESS Equations and Modeling 6 Two truckers start on a trip at the same time. One heads west at 60 mph while the other heads east at 55 mph. Their CB radios work for a distance up to 500 miles apart. How long will they be able to communicate with each other? Model and solve algebraically. Solution: 4 hr 20 min. MATHEMATICS for COLLEGE READINESS Equations and Modeling 7 Solve the system of equations algebraically. 3x 4 y 2 y x 1 3 4 6 Solution: lines are parallel. empty set: The MATHEMATICS for COLLEGE READINESS Equations and Modeling 8 Solve for w: p = 2 ( l + w ) Give your answer in two forms, one answer with a single fraction and the other as a sum or difference of two terms. Solutions: p 2l w 2 p w l 2 MATHEMATICS for COLLEGE READINESS Equations and Modeling 9 1 1 3 Solve for a: a b Solution: b a 1 3b MATHEMATICS for COLLEGE READINESS Equations and Modeling 10 Solve for x: 1 1 1 x y a ay Solution: x ya MATHEMATICS for COLLEGE READINESS Equations and Modeling 11 The Boosters Club held a spaghetti dinner as a fundraiser. They sold 300 tickets and collected $2200. If an adult’s ticket cost $8.50 and a child’s ticket cost $3.50, how many of each were sold? Solution: 230 adult tickets 70 children’s tickets MATHEMATICS for COLLEGE READINESS Equations and Modeling 12 Admission to a school football game was $2 for students and $3 for nonstudents. How many of each group attended if there were 900 people and $1920 collected? Model and solve algebraically. Solution: 120 nonstudents. 780 students. MATHEMATICS for COLLEGE READINESS Equations and Modeling 13 Solve algebraically: x2 – 5 = 3x Solutions: 3 29 x 2 MATHEMATICS for COLLEGE READINESS Equations and Modeling 14 Bill needs to make 1000 copies before the meeting that starts in 15 minutes. He begins using a machine that makes 30 copies per minute after laminates, a newer machine which copies at 45 pages per minutes become free . Using both machines, will Bill be able to make all the copies before the meeting? Explain algebraically. Solution: No, it will take almost 17 min to make the copies. MATHEMATICS for COLLEGE READINESS Equations and Modeling 15 Find the equation of a line with slope of 0 through (6, 10). Solution: y = 10 MATHEMATICS for COLLEGE READINESS Function Theory 16 Which of the following are functions? a. x3 y = 12 e. x y 3x 4 2 b. x2 + y2 = 20 f. y(y + 2) = 4 + x c. x3 + y = 15 g. d. x (x+2) = 5 + y h. y = 5 Solution: c, d, e, h MATHEMATICS for COLLEGE READINESS x=4 Function Theory 17 If f(x) = x2 – 5, find f(b) – f( c) . Solution: b2 – c2 MATHEMATICS for COLLEGE READINESS Function Theory 18 If f(x) = 2x2 – x + 3, find f(x + h) . Solution: 2 x2 4 xh 2h2 x h 3 MATHEMATICS for COLLEGE READINESS Function Theory 19 If f(x) = x2 – 4, find f(a + 5) . Solution: a2 +10a+21 MATHEMATICS for COLLEGE READINESS Function Theory 20 If f(x) = x2 – 3 x + 5, find f(x + h) – f(x) . Solution: 2xh + h2 – 3h MATHEMATICS for COLLEGE READINESS Function Theory 21 If f(x) = 3x4 – 4x3 -2x2 +x - 5, find f(-x). Solution: 2xh + h2 – 3h MATHEMATICS for COLLEGE READINESS Expressions with Exponents 22 Simplify: x y y 1 Solution: 0 2 1 1 y 3 MATHEMATICS for COLLEGE READINESS Expressions with Exponents 23 Simplify. Give the answer with positive exponents only: 3 2 3 4 x x 0 Solution: 8 x4 MATHEMATICS for COLLEGE READINESS Expressions with Exponents 24 Simplify. Give the answer with positive exponents only: 6 x 2 y 3 z 4 . 1 2 2 4x y z 3 y5 . Solution: 6 2x z MATHEMATICS for COLLEGE READINESS Expressions with Exponents 25 Multiply 2 5 4 8r s 3r s a. 5r 7 s5 b. 5r10s 4 c. 7 5 24r s d. e. 24r10s4 11r 7 s 5 7 5 Solution: d. 24r s MATHEMATICS for COLLEGE READINESS Expressions with Exponents 26 Multiply 9r Solution: 9r 1/2 s r 9/10 5/4 s MATHEMATICS for COLLEGE READINESS 2/5 1/4 s Expressions with Exponents 27 Multiply 16r Solution: 1/2 s r 3/4 16r s MATHEMATICS for COLLEGE READINESS 5/4 s 0 Expressions with Exponents 28 If the area of a circle if r 2 , what is the area of a circle if the radius is 3a? Solution: A 9 a 2 MATHEMATICS for COLLEGE READINESS Expressions with Exponents 29 4 If the volume of a sphere is r 3 , what is the volume of a sphere with radius 3a? 3 Solution: V 36 a3 MATHEMATICS for COLLEGE READINESS Expressions with Exponents 30 If the volume of a cylinder is r 2 h, find the volume of a cylinder with radius 3x and a height that is twice the radius. Solution: V 54 a3 MATHEMATICS for COLLEGE READINESS Exponential and Logarithmic Functions 31 If t e x 3, a. 3 + ln t b. t3 e then x = c. Solution: e. ln t – 3 MATHEMATICS for COLLEGE READINESS t 3 d. ln ( t – 3 ) e. ln t – 3 Exponential and Logarithmic Functions 32 If 7x = 3, then x = a. 3 7 b. 7 3 Solution: c. log 3 7 d. log7 3 e. 7 log10 3 c. log 3 7 MATHEMATICS for COLLEGE READINESS Simplifying Expressions 33 Simplify 3x 2 x x 3 x 4 a. 21x 4 x 2 b. 27 x 4 x 2 c. 10 x 2 20 x d. 10 x 2 10 x Solution: a MATHEMATICS for COLLEGE READINESS Simplifying Expressions 34 Simplify 2a 2 b a 2b 2 2 a. a 2 3b 2 b. a 2 5b 2 2 2 c. a 4ab 5b 2 2 a 4 ab 3 b d. Solution: c MATHEMATICS for COLLEGE READINESS Simplifying Expressions 35 Simplify 30k 7 k 4 5k Solution: 5k2 + 250k -16 MATHEMATICS for COLLEGE READINESS 2 Simplifying Expressions 36 Evaluate the following expression using x = -3.1 and y = 4.23 4x-2 + y3 Solution: 76.103 MATHEMATICS for COLLEGE READINESS Algebraic Expressions 37 Factor completely using integer coefficients. 12m4 n3 48m2 n3 Solution: 12 m2n3 (m2 + 4) MATHEMATICS for COLLEGE READINESS Algebraic Expressions 38 Factor completely using integer coefficients. 12m2 8m 9m 6 Solution: (4m – 3) (3m + 2) MATHEMATICS for COLLEGE READINESS Algebraic Expressions 39 Which of the following is a factor of 12 x 2 x 6? a. b. 3x 2 4 x 3 4 x 3 c. d. not factorable Solution: c. (4x + 3) MATHEMATICS for COLLEGE READINESS Algebraic Expressions 40 Which of the following is a factor of 3x3 18 x 2 48 x ? a. x 4 b. x 2 c. x 8 d. x 2 Solution: b. (x + 2) MATHEMATICS for COLLEGE READINESS Algebraic Expressions 41 Write in simplest radical form. 180x9 y 8 Solution: 6 x4 y 4 5x MATHEMATICS for COLLEGE READINESS Algebraic Expressions 42 Write in simplest radical form. 49x 4 y 2 Solution: 7x2y MATHEMATICS for COLLEGE READINESS Algebraic Expressions 43 Write in simplest radical form. 2 2 3 Solution: 2 3 2 MATHEMATICS for COLLEGE READINESS Algebraic Expressions 44 Write in simplest radical form. 3 32 7 50 Solution: 23 2 MATHEMATICS for COLLEGE READINESS Algebraic Expressions 45 Perform the indicated operations and reduce answers to lowest terms. m n m 2 mn m n 2 a. m 2 m 2mn n 2 m n m n mn b. mm n c. m 1 d. m MATHEMATICS for COLLEGE READINESS Solution: d Quadratic Equations 46 Solve algebraically. 25 x 2 144 0 12 12 , Solution: x 5 5 MATHEMATICS for COLLEGE READINESS Quadratic Equations 47 Solve algebraically. 6 x 2 150 Solution: x 5,5 MATHEMATICS for COLLEGE READINESS Quadratic Equations 48 Solve algebraically. 6 6x 5 x 2 3 , Solution: x 3 2 MATHEMATICS for COLLEGE READINESS Quadratic Equations 49 Solve algebraically. x2 x 4 1 17 Solution: x 2 MATHEMATICS for COLLEGE READINESS Quadratic Equations 50 Solve algebraically. 3x 2 2 5 x Solution: x 1 ,2 3 MATHEMATICS for COLLEGE READINESS Rational Expressions 51 State the restricted values: Solution: x ≠ –3, 2 MATHEMATICS for COLLEGE READINESS x 1 1 2 x2 x 6 Rational Expressions 52 Over the set of real numbers, state where the function is undefined: x2 4 x2 9 Solution: The expression is undefined over the set of real numbers where -2<x<2 or x = -3 or 3. MATHEMATICS for COLLEGE READINESS Rational Expressions 53 x2 4 x2 5x 6 Find the product: 3 x 6 4 x 12 x 2 2 Solution: 12 MATHEMATICS for COLLEGE READINESS Rational Expressions 54 Calculate and write the answer in scientific notation. 85 10 0.03 10 0.5 10 3.2 10 4 1 a. 1.59375 1011 b. 1.59375 1011 c. 1.59375 101 d. 1.59375 109 Solution: a MATHEMATICS for COLLEGE READINESS 2 8 Rational Expressions 55 x2 y 2 , Find the area of a rectangle if the length is y x y and the width is . x y Solution: ( x y)2 y MATHEMATICS for COLLEGE READINESS Rational Expressions 56 5x 5 Add and simplify x 2 1 x 2 1 . 5 Solution: x 1 MATHEMATICS for COLLEGE READINESS Rational Expressions 57 3a Simplify 2b . 2b 5a Solution: 3 5 MATHEMATICS for COLLEGE READINESS Rational Expressions 58 3 4 Simplify x . x 1 x2 3x 4 x 2 Solution: x 1 MATHEMATICS for COLLEGE READINESS Rational Expressions 59 x3 y 3 x 2 y 2 Simplify: 2 x 2x 2 x 2 2 xy 2 y 2 Solution: x 2 xy MATHEMATICS for COLLEGE READINESS Rational Expressions 60 Simplify: x x y 6 x y 3 18 a. x x y x y b. 2x c. 2x x y x y 6x d. 3 MATHEMATICS for COLLEGE READINESS Solution: b Rational Expressions 61 Simplify: 3 a. 2 x 1 3 x2 x 2 2 2 x 4 x 1 3 b. 2 x 1 c. 3 x 2 2 x 2 x 1 3 d. 2 MATHEMATICS for COLLEGE READINESS Solution: a Rational Expressions 62 5v 5 v2 1 Simplify: v 2 v 2 4v 4 a. b. 5 v 2 v 1 5 v 1 v 2 v 1 v 1 Solution: d MATHEMATICS for COLLEGE READINESS c. d. 5 v 5 v 2 v 1 v 1 5v 2 v 1 Rational Expressions 63 What values for x are not allowed? 3x 4 x 5 x2 2 x 8 5 a. 0, 4 5 c. 4,0, , 2 4 b. 4,2 d. 4, , 2 Solution: b MATHEMATICS for COLLEGE READINESS 5 4 Miscellaneous 64 Which of the following sets of numbers could NOT be the measures of the sides of a right triangle? a. 3,7, 10 c. 6,8,10 b. 8,5 3, 11 d. 10,24,26 Solution: a MATHEMATICS for COLLEGE READINESS Miscellaneous 65 Find the length of the hypotenuse of the right triangle whose legs are 5 and 11. Solution: 6 MATHEMATICS for COLLEGE READINESS Miscellaneous 66 Find the distance between (-3,5) and (-7,8). Solution: 5 MATHEMATICS for COLLEGE READINESS Miscellaneous 67 Find the coordinates of the point midway between (-5,2) and (3,-10). a. (-4,6) b. (-1,-4) c. (1,6) d. (-4,-1) Solution: b. (-1, -4) MATHEMATICS for COLLEGE READINESS Miscellaneous 68 Find y in the diagram below. a. 24 b. 18 c. 4 d. 8 9 6 12 Solution: a MATHEMATICS for COLLEGE READINESS y Miscellaneous 69 Find y in the diagram below. 12 8 y Solution: 7.5 MATHEMATICS for COLLEGE READINESS 15 Variation 70 Find an equation of variation where y varies inversely as x, and y 10 when x 3.5 a. b. c. d. y 35 x 3 y x y 6.5 x 35 y x Solution: d MATHEMATICS for COLLEGE READINESS Variation 71 Y varies directly as the cube root of x. Y is 30 when x is 1. Find Y when x is 27. a. b. c. d. 10 90 15 30 Solution: b. 90 MATHEMATICS for COLLEGE READINESS Variation 72 The number of hours (H) it takes to do the job in inversely proportional to the number of people (P) working on it. It takes 4 people 12 hours to do the job. How long would it take 6 people to do the job? a. b. c. d. 18 hours 10 hours 8.5 hours 8 hours Solution: d MATHEMATICS for COLLEGE READINESS Content Guide Equations and Modeling Function Theory Expressions with Exponents Exponential and Logarithmic Functions Simplifying Expressions Algebraic Expressions Quadratic Equations Rational Expressions Miscellaneous Variation MATHEMATICS for COLLEGE READINESS Slides 1 – 15 Slides 16 – 21 Slides 22 – 30 Slides 31 – 32 Slides 33 – 36 Slides 37 – 45 Slides 45 – 50 Slides 51 – 63 Slides 64 – 69 Slides 70 – 72