MAT 15 In Class Assignments 1
Chapter 1 1. Simplify each expression: a) 5 b) (5 ) c) 4 d )0 6 4. a)factor 4,56 into the product of prime factors b) Reduce 4,56 5,148 to lowest terms.. Translate each statement into an equivalent inequality: a) x is at least 4 b) x is between 0 and 8 c) x is no more than 5 4. Which property or properties of the real numbers is illustrated by each expression? a) x x b)5 0 5 c)5 ( 5) 0 d)5 x( y) x(5 y) e)(1) f )( x y) x y 5. Write in scientific notation: a) 76,000 b) 4,50 c) 0.0000067 6. Write in expanded form: 5 4 a)1.94x10 b)5.4x10 c ).5x10 7. Simplify each expression. Write all answers with positive exponents only. Assume all variables are nonzero. 4 4-5 4 - ( ) ( ) )( y ) )( ) (5y ) ) x x 10 a x b y c x
Section.1 Solving Linear Equations 1. Solve the following equations: 5 a) 4a 8 a 7 b) t 1 1 1 5 c)7 ( x ) 4( x 1) d) x x 4 4 e) x (x ) 6 f )x (x 5) 6. The taxi meter was invented in 1891 by Wilhelm Bruhn. Chicago charges $1.80 plus $0.40 per mile for a taxi ride. a) If the fare is $6.60, write an equation that shows how the fare is calculated. b) How many miles were traveled if the fare was $6.60?. In 199, twice as many people visited their doctor because of a cough than an earache. The total number of doctor s visits for these two ailments was reported to be 45 million. a) Let x represent the number of earaches reported in 199. Write an expression for the number of coughs reported in 199. b) Write an equation that related 45 million to the variable x. c) How many people visited their doctor in 199 to report an earache?
Section. Solving Linear Inequalities 1. Solve the inequalities, graph each solution, and write the solution in interval notation: 1 a) y>4 b) 1 x 1 c)5x 115 d)10 y 6 e)(x 1) 10 f )1 4(a 1). A store selling art supplies finds that they can sell x sketch pads each week at a price of p dollars each according to the formula x=900-00p. What price should they charge is they want to sell: a) at least 00 pads each week? b) fewer than 55 pads each week? c) more than 600 pads each week? d) at most 75 pads each week? 4
Section.4 Solving Compound Inequalities 1. Solve the following inequalities. Use a line graph and interval notation to write each solution set. a) 60 0a 0 60 b)7x 5 or x 1 c)5 x+1<9 d)8 1 x and x>1 4 e)x 1 x 4 or 5x x 4 f) m 5. A factory s quality control department randomly selects a sample of 5 lightbulbs to test. In order to meet quality control standards, the lightbulbs in the sample must last an average of at least 950 hours. Four of the selected bulbs lasted 95 hours, 1000 hours, 950 hours, and 900 hours. How many hours must the fifth lightbulb last for the sample to meet quality control standards?. A worker earns $1 per hour plus $16 overtime pay for every hour over 40 hours. How many hours of overtime are needed to make between $600 and $800 per week? 5
Section.1 The Rectangular Coordinate System 1. Identify the quadrant in which each point is located, and place it on the graph below. Label the axes on the graph. a) (1,) b) (-1,-) c) (5,0) d) (0,) e) (-5,-5) f) (1/,) g)(-8,) h) ( 6,-). The x-axis on the graph below represents number of hours worked and the y-axis represents pay, in dollars. Approximately what is the pay for working 10, 0, 0, and 40 hours? Create a table that displays the data shown on the graph. 6
Section. Slope 1. Find the slope of the line from the given graphs:. Find the slope of the line through the given points. Then plot each pair of points and draw the line through them. a) (,1) (4,4) b) (, -5) (, -) c) (1,5) (-4,-5). A line, l, contains the points (,4) and (-,1). Give the slope of any line perpendicular to l. 4. Determine if the line through the points (7,) and (-9,) and the line through the points (4,-4) and (1, -4) are parallel, perpendicular, or neither. 5. Solve for y: y x b 0 m 7
Section. Graphing Linear Equations 1. Find the slope, x-intercept, and y-intercept of each equation. Sketch the graph using this information. a) y x b)x y 1 c)4x 5y 0 d) y x e) y x 4. Give the slope and y-intercept of y = -.. For the line x = -, sketch the graph, give the slope, and find any intercepts. 4. Find the x- and y-intercepts of the graph of the equation 4x 16. 5. Find the equation of a line with an x-intercept of (,0) and a y-intercept of (0,). 6. In an aerobics class, the instructor tells her students that exercise heart rate is 60% of the maximum heart rate, where maximum heart rate is 0 minus their age. Determine the equation that gives exercise heart rate, E, in terms of age, A. Find the exercise heart rate of a year old student. 8
Section.4 More on Graphing Linear Equations 1. Complete the table: Equation Slope,m x-intercept y-intercept y x 5 y x 4x 6y 4 y 1 x. Write each equation in slope-intercept form: a)6x 4y 1 b) y 1 ( x 5) c)4x y 0. Find the equation of a line with slope - that passes through the point (0,7). 4. Find the equation of a line parallel to y=5x-1 and has x-intercept (4,0). 5. What is the equation of a line perpendicular to the graph of y=x and passes through ( -,5)? 6. What is the equation of a line that passes through (,1) and (1,)? 7. A salesperson earns a salary of $1500 per month plus a % commission of the total monthly sales. Write a linear equation giving the salesperson s total monthly income I in terms of the sales, s. Sketch a graph of the equation, clearly labeling two points on the graph. 9
Section.6 Introduction to Functions 1. For each of the following relations, give the domain and range, and state which are functions: a){(1,)(,5)(4,1)} b){(5, )(, )(5, 1)} c ){(,1)(5,7)(,)}. Let f ( x) x 5 and g( x) x x 4. Evaluate the following: a) f () b) f ( ) c) g( ) d) g(0) e) f () g() f ) g( t) g) f ( a). Graph each function and identify its domain and range: a) f ( x) x 1 b) g( x) x + c f x x d g x ) ( ),x 0 ) ( ) 4. The function V ( t) 00t 18, 000 where V is value and t is time in years can be used to find the value of a large copy machine during the first 5 years of use. a) What is the value of the copies after years and 9 months? b) What is the salvage value of the copier if it is replaced after 5 years? c) State the domain of this function. d) Sketch a graph of this function, clearly labeling values and the axes of the graph. e) What is the range of this function? f) After how many years will the copier be worth only $10,000? 10
Section 4. Solving Systems of Linear Equations Algebraically 1. Solve the following systems: x y 4 a 5b 7 a) b) x y 6a 5b 9x 5y 9 p 5q c) d) 9x 5y 9 4 p 10q a b 6 e) f) a b 8 x 5 4 x 5 y y 1 1 x 4y 7 g) 6 x y 5. A store sells sheets for either $15 or $0. One month, sales totaled $1,570. If the store sold 56 sheets, how many $15 sheets were sold?. The difference of the measures of two complementary angles is 6 degrees. Find the measure of each angle. (Complementary angles add up to 90 degrees). 11
Section 5.1 Addition and Subtraction of Polynomials 1. Evaluate each polynomial for the given value of the variable: a x x x x ) 7 6 b x x x x )8 1 0. Add or subtract as indicated: a) Subtract 4 4 6a 9a a 5a from 10a 1a a 7 b) Add 4 a 7a a 8a 10, c) (x 5) (x 5) d) 6 a { a 6[a ( a 1) 6]} e) f) 8a 1, and 10 9 7 4 a a a. (11 x xy y ) (9x xy y ) ( 6x xy 5 y ) (m n mn 7) ( mn ) ( m n 7). The calculated annual fixed cost (in dollars) of owning and operating an automobile is given by the polynomial 0x 45x 5400, where x is the number of years after 1999. The variable costs are represented by the polynomial 75x 0x 108x 1550. Find a polynomial that represents the difference between the fixed cost and the variable cost for a given year. 1
Section 5. Multiplication of Polynomials 1. Multiply as indicated: a x x x ) (6 5 4) b a b a ab b ) ( ) c)( x 4)( x 6) d x x )( )( 5) e x y x xy y )( )(4 ) f )(5 t)(4 t) g)(4x 1) h)( x 1)( x )( x ) i b 4 )( 1)( a 5) j)(t 10)(t 10) 1 k)( q p). For (a) and (b) below, find f ( x) g( x ), f(a+), and g(x+1)-g(x): a) f ( x) x and g( x) x 1 b) f ( x) x x and g( x) x x 1
Section 5. Division of Polynomials 1. Divide as indicated: 1m 15m 9y y y a) b) 4 m y 4 6 5 4 0a b 4ab 6x y 18x y 9x y c) d) 8ab 6x y 4 4 5 x x 7 10x x e) f) x x g x x x )(56 ) (8 1) h x x x )( 6 40) ( 10) i x x 4 )( 5) ( 5). Find f( x) gx ( ) : a f x x x g x x ) ( ) 5 4; ( ) 8 b f x y g x y 4 ) ( ) 16; ( ) c f x x g x x ) ( ) 8; ( ). To find the average cost of producing an item, divide the total cost by the number of items produced. A company that manufactures computer disks uses the function C(x)=00+x to represent the cost of producing x disks. a) Find the average cost. b) What happens to the average cost as more items are produced? Hint: Set up a table. 14
Section 5.4 GCF and Factoring by Grouping 1. Factor out the greatest common factor: a)4x 16x 0 x b) x y xy x y c x a b y a b d x x y x x y )5 ( ) ( ) )10 ( ) 15 ( ) e)0a b c 0ab c 5 a bc f ) ax x bx ab g)9x 18x 4x 8 h) y( x ) ( x). Solve for P : TM=PC+PL. In a polygon with n sides, the interior angles, measured in degrees, add up to 180n -60. Find an equivalent expression by factoring out the GCF. 4. The area (in square meters) of a pool is given by the expression is the length of the pool. A l 0l, where l a) Factor the expression for the area. b) The width of this pool is 0 m. What is the length? 15
Section 5.5 Factoring Trinomials 1. Factor: a y y b a a ) 6 ) 6 c)x 14x 0 x d) x 10xb 5b e) m 8mn 9 n f ) 7a 6a g)6x x x 4 h)60 p 8p q 16 pq i x xy y j x x x x x ) 1 70 49 ) ( 5) 7 ( 5) 6( 5). Find the missing factor: a x x x )1 8 ( )(6 ) b x xa a x a ) 48 ( 6 )( ) c x x x ) 15 ( )( ). Factor, if possible: a y xy y b a a )1 17 )9 6 1 c)x 4x 96 x d) x 5xb 6b e) m mn n f )9 6a a g t ) 6t -8 16
Section 5.6 Special Factoring 1. Factor, if possible: a)5 10t t bx ) 100 cx d 4 ) 81 )( x ) 9 ea ) 8 f )6 60x 5x g) x 144y 4 h)7m 6m 1m i x y a x y )16( ) ( ) j)50xy 18x y k 4 )9x y 81 l)9a 1ab 4b m)9 60 pq 100 p q. The volume of a box can be modeled by the polynomial expression completely. 7x 4x x. Factor this 17
Section 5.7 Solving Quadratic Equations by Factoring 1. Solve each equation: a t t )9 1 0 b x x ) 4 0 c)16x 5x d)800x 100x e x )( 1) x 7 f x x ) 5 0 g x x )4 6 h) n( n 7) 4 i)( n )( n 4) 1n j x ) 8x 1 0. Find all values such that f ( x) g( x ) : a f x v v g x ) ( ) 10 5 ; ( ) 0 b) f ( x) 4 y y; g( x) 1 y c) f ( x) t 18; g( x) t. A ball is dropped from a balloon 900 feet above the ground. The height, h, of the ball above the ground (in feet) after t seconds is given by the equation h 900 16t. When will the ball hit the ground? 18
Section 6.1 Multiplication and Division of Rational Expressions 1. Identify the values for which the given expression is undefined: 1 a) x x b) t 5 6t. Perform the indicated operation and express the answers in lowest terms: x 9 x- x 5x 1 x 1 b x x x x a) ) 4 x- 4 4 5 1 c p q q 6p q ) p pq p pq q p d) a 7a 1 a 9a 18 a 5 a 7a 10 e 4t 1 8t 1 ) 6 7 t t t 8 f ) 1a b ab 4b 6a 15 9a 6b ab 8a b b 4 6b g a 16b a 9ab 0b a 5b ) a 8 ab 16 b a 7 ab 1 b a 6 ab 9 b, For f ( x) and g( x ), find f ( x) g( x) and f( x) gx ( ) : y y y a) f ( x) ; g( x) y y 6 y 1 1 5 6 m 7m 1 9 m b) f ( x) ; g( x) m m 4 19
Section 6. Addition and Subtraction of Rational Expressions 1. Perform the indicated operation and simplify where possible: 8 5 7 a) b) x x p q q p n n 4x x c) d) 4n 4 n 1 x 1 x 4 1 18y x 8 x e) f) y 8y 7 x 8x 4 x 5x 6x 1 x 4x y y 4 y g) h) x 1 x 1 x x 1 8y 16 y ( y ). Given f ( x) and g( x ) f ( x) g( x) and f ( x) g( x ) : 9 6 a) f ( x) ; g( x) 9x 6x 8 9x 4 1 ab b) f ( x) ; g( x) a b a b. The formula 1 1 P a b is used by optometrists to determine how strong to make eyeglasses. If a =10 and b =0., find the value of P. 4. Write an expression for the sum of the reciprocals of two consecutive integers and simplify it. 0
Section 6.4 Solving Rational Equations 1. Solve the following equations: 5 1 x a) b) x x 1 x x x 4 y 6 c) d) 0 x x 9 x y y y 1 n 1 n 1 1 e) n n n n. If f( x) 4, find all values for which f(x)=. x 1 pv 1 1 pv. Solve for V T T 1 1 4. If f ( t) and g( t), find all values for which f(t)=g(t). t 1 t 1 5. At a warehouse, it takes two employees 1 1 hours to load a truck when they work together. If it takes one employee working alone hours to load a truck, how long does it take the other employee working alone? 1
Section 7.1 Radical Expressions and Radical Exponents 1. Find the value of each root: a) 144 b) 100 c) 0.04 d ) 7. Simplify each expression: a) 49 a b) x y 10 6 10 c) ( x ) d) 8a 15 1 5 e) f) 81 5 1 g x y h 16 5 5 5 )( ) ) 81 4 y 4 4 5 i)( t ) 10 5 6 p j) 7 p. The time,t, it takes in seconds for an object to fall d feet is given by the equation 1 t d. 4 a) The Sears Tower in Chicago is 1450 feet tall. How long would it take a penny to fall from the ground from the top of the Sears Tower?
Section 7. Simplifying Radical Expressions 1. Write each of the expressions in simplified form: a. 4 d. b. 7 e. 49 100 54xy 6y c. 4 7 4v 40x y f. 4 81u d. 5 48a b c g. 5 p qr 5 10. Find the distance between (-5,4) and (,-).. Evaluate and simplify b 4ac if a =, b = -6, and c =. 4. Show that a b a b is not true by using a = 9 and b = 16 and simplifying both sides. 5. The length of a diagonal of a rectangular box with length l, width w, and height h is given by d l w h. Find the length of the diagonal of a rectangular box that is feet wide, 4 feet long, and 1 feet high. 6. Simplify x x 16 64
Section 7. Addition and Subtraction of Radical Expressions 1. Combine the following expressions, if possible. Assume that any variables under an even root are nonnegative. a. 4 f. x y 7x xy 4 b. 6y a y a g. 8 6 8 x y y 8x c. 5x 6 x 6 x 6 h. 5 5a 7ab 6b 1a b 5 16 4 54 i. 7r 4r 64r d. 6 e. 5 y y j. 1 18 1 5. Find f ( x) g( x) and f ( x) g( x ) : a. f ( x) 81 and g( x ) 75 b. 5 f ( x) 0 a b and g( x) 45a. The ramp shown at the right has a base, b, of 8 feet and a height, h, of 4 feet. The length of the ramp is given by Find the length of the ramp. b h. 4
Section 7.4 Multiplication and Division of Radical Expressions 1. Multiply or divide as indicated and simplify. Assume that any variables under an even root are nonnegative. a. 5 7 b. 6 9 c. x 5 x d. x e. a b f. v 7 v 7 g. 5 v 5 v h. i. j. x x 9b 6 a 6 a (rationalize the denominator). Find f ( x) g( x) and f( x) gx ( ) if f ( x) x 1 and g( x) x. 5
Section 7.6 Complex Numbers 1. Simplify the following as much as possible: a) 6 b) 49 c) 48 d) 1 e) 65 f ) 9 49 g) 4 49 h) 9 4 i) 7 j) 6 48. ( +4i)+(-i)=. (4-9i)(+I )= 4. 4 4 i = 6
Section 8.1 Solving Quadratic Equations by Completing the Square 1. Solve by completing the square: a x ) x 8 0 b x ) 4x 0 c x x ) 9 1 0 d m ) 7m 0 e t t ) 6 1 f x x ) 4 8 0 g x x )4 5 0. Given f ( x) and g( x ), find all values for which f(x)=g(x): a) f ( x) x +7x; g( x) 6x 8 b) f ( x) x - x; g( x) x 15. A rectangle is 4 feet longer than it is wide, and its area is 0 square feet. Find the dimensions of the rectangle, to the nearest tenth of a foot. 4. If the lengths of the shorter sides of a right triangle are both x, find the length of the hypotenuse, in terms of x. What are the measures of the angles in this triangle? x x 7
Section 8. Solving Quadratic Equations by Using the Quadratic Formula 1. Solve using the quadratic formula: a x x ) 4 0 b x x ) 6 0 r r 1 c) 4 5 10 d m ) 4m 1 0 e)t t t f x x ) 10 18 x 5x x g) 6. Use the discriminant to determine the number and types of solutions for each of the following equations: a)x 4x b)x 1 6x r 5r c) 1 d x ) 6m 10 0 e t t ) 6 1. A woman invests $1,000 in a fund. Interest is compounded annually at a rate r. After one year, she deposits an additional $,000. After two years, the balance in the account is $,68.10. This is calculated as follows: Find the rate, r.,68.10 1,000(1 r),000(1 r ) 8
Section 8.4 Graphing Quadratic Functions 1. Answer the following questions for the graph shown: a) What are the x-intercepts of the graph? b) What is the y-intercept? c) What is the axis of symmetry? d) What is the vertex? e) What is the domain and range?. For the following functions, find the x-intercepts, y-intercept, axis of symmetry, vertex, determine if the graph opens upward or downward, and find any maximum or minimum values. A sketch of the graph is helpful. a y x x ) 6 5 b y x x ) 4 4 c) f ( x) x 4x d f x x x ) ( ) 4 1 9 9
Some problems in this packet have been taken from or adapted from the following texts: Akst and Bragg, Introductory Algebra Through Applications, Seventh Edition Akst and Bragg, Intermediate Algebra Through Applications McKeague, Charles, Intermediate Algebra, Seventh Edition. Tussy and Gustafson, Elementary and Intermediate Algebra, Third Edition 0