Algebra Non-Calculator Skills Inventory Solutions

Transcription

1 Algebra Non-Calculator Skills Inventory Solutions 1. True or False: If a and b are non-zero real numbers then = + a + b a b Solution: False. Example: If a = 4 and b = 1, then = b + a The following statement is true: + = a b ab. True or False: If c and d are non-zero real numbers then Solution: False. Example: If a = 1, b =, c =, d = 4, then a + b 1+ 1 = = + c + d True or False: If c is a non-zero real numbers then Solution: True 4. True or False: If b and c are non-zero real numbers then a a a = + b + c b c Solution: False See #1 5. True or False: If 10u + v and u + v are nonzero, then 10 t + u t + u = 10u + v u + v Solution: False. Example: Let t =, u = 1, and v =, then 10t + u 10() = = 10u + v 10(1)

2 6. True or False: 4 = 9 Solution: False. 7. True or False: If a and b are non-negative real numbers then a b = ab Solution: True 8. True or False: 16 = ± 4 Solution: False The symbol denotes positive square root. 9. True or False: Solution: False ( ) a + b = a + ab + b 10. True or False: If a and b are non-negative real numbers then a + b = a + b Solution: False Example: If a = and b = 4, then + 4 = = 5 = True or False: If b is a non-zero real numbers then a b = a b Solution: True 1. True or False: If a and b are non-negative real numbers then a + b = a + b Solution: False Example: If a = 4 and b = 9, then =

3 1. True or False: = 9 Solution: False Note that the negative sign is not being squared so - = True or False: = 16 = 7 Solution: False ( ) ( ) 15. True or False: ( a b) = a + b Solution: True 16. True or False: Solution: False Example: Let a = and b = -. x = x 17. True or False: ( ) Solution: False (x) = 4x 18. True or False: = 9 Solution: False = True or False: If ac < bc, then a < b. Solution: False If c is negative, the inequality sign reverses. 0. True or False: The slope of the line x = 0 is 0. Solution: False The slope of the line is undefined.

4 Multiple Choice 1. Which equation has no solution? A. x + 4 = (x + 4) B. x + 4 = (x + 1) C. x + 5 = (x + ) D. Neither A nor B has solutions. Solution: D A. x + 4 = (x + 4) B. x + 4 = (x + 1) x + 4 = x + 8 x + 4 = x + 4 = 8 4 = no solution no solution C. x + 5 = (x + ) x + 5 = x + x = -. Solve for x: y = mx + b A. B. C. D. y b x = m b + y x = m y x = + b m b x = y m y b Solution: A y = mx + b y b = mx = x m

5 . What system is graphed at the right? A. x + y = y = -x - 1 B. x + y = x + y = -1 C. x + y = 6x + y = 9 D. x + y = y = -x - 1 Solution: D The slope of the two lines is - and the intercepts are and -1 A. x + y = y = -x + y = - ½ x + NO y = -x - 1 B. x + y = y = -x + y = - ½ x + NO x + y = -1 C. x + y = y = -x + YES 6x + y = 9 y = -6x + 9 y = -x + Same line D. x + y = y = -x + YES y = -x - 1 YES

6 4. Which system is graphed at the right? A. x + 4 y 4 4x + y y x 4 B. 1 y x y x 4 C. 1 y x D. x + 4 y 4 4x + y Solution: B The slope of one line is ½ and the slope of the other is. The shading is a above the line with a slope of ½ and below the line that has a slope of. 5. Find the x-coordinate of the solution to the system: x + y = 1 5x = y + 11 A. 1 B. C. D. 5 Solution: C In the second equation, subtract y from both sides and then add the two equations. x + y = 1 5x - y = 11 8x = 4 x = 6. What is the value of A. B. 4 C. 7 D. 16 c c 1 c 4 when c =? Solution: D c c 81 9 = = = = c

7 7. Simplify the expression ( 5x y )( xy ) A. B. C. D. 5 5x y 5 5y x 5 y 5x 5xy y Solution: B ( 5x y )( xy ) = 5x y = 5x y = x 6 a b 8. Simplify the expression 5 18a b b A. 6a 4 B. 6a b 4 b C. 6a 4 b D. 6a Solution: D a b b b = = 18a b 6a 6a In simplified form ( ) is 1 A. 7 1 B. 7 C. -7 D. 7 Solution: A ( ) 1 1 = = 7 ( )

8 0. Simplify ( x 5 x + x ) ( x 5x + 7) A. B. C. D. 5 x x x x x + 8x 10 5 x + x 8x x x x + 4 Solution: B ( x 5 x x ) ( x 5x 7) + + = 5 x x + x x + 5x 7 = 5 x x + 8x ( 4 p ) = A. B. C. D. 16 p 6 p p 1 p p 4 p p 1 p 9 Solution: C. When the expression 6x x 1 + is factored completely, one of the factors is A. x - 4 B. x + C. x + 4 D. 6x + 1 Solution: A 6x + x 1 = (x 4)(x + ) = A. 0 B. 5 6 C. 6 6 D. 1 6 Solution: B = = = 5 6

9 4. Which point is not in the graph of the inequality x y 1. A. (, 1) B. (1, ) C. (-1, -1) D. (0, -) Solution: C Substitute () - 1 = 5 >1 (1) - = 1 > 1 (-1) - (-1) = - < 1 (0) - (-) = > 1 5. Find the domain and range of the function y = 5 + x A. domain: all reals; range: y 0 B. domain: all reals; range: y 0 C. domain: x 5 ; range: y 0 D. domain: x 5 ; range: y 5 Solution: A It doesn't matter what you substitute for x so the domain is all real numbers. The absolute value gives only positive numbers or 0 so the range is y 0

10 Problems 6-9 refer to the graph at the right. The graph shows the cost of renting skates from each of two different places. 6. Suzie's Skate Land requires no deposit. Find the rate of change, or cost per hour. A. $1/hour B. $/hour C. $/hour D. $6/hour Solution: B Suzie's Scott's 7. Scott's Skates requires a non-refundable deposit, and the first hour is free. What is the amount of the deposit? A. $1 B. $ C. $ D. $6 Solution: C 8. Maria wants to rent roller blades for four hours. At which shop will the rental be less expensive? A. Suzie's Skate Land B. Scott's Skates C. The cost is the same. D. Answer cannot be determined from the graph. Solution: B 9. For what rental time will the skate rentals cost the same? A. 1 hour B. hours C. hours D. 6 hours Solution: C

11 40. Given f ( x) = x 5, find f (1) f (5) A. 0 B. -4 C. 14 D. -14 Solution: A f f ( ) ( ) (1) (5) = = ( ) = What is the value of A. - B. does not exist C. 1 D. 1 4? Solution: A 4. The rate of change is constant in the graph. Find the rate of change. Explain what the rate of change means for the situation. A. 0, value drops $0 every year B. 60, value drops $60 every year C., value drops $00 every years D. 00, value drops $00 every year Solution: C Rate of change is the slope. The slope of the line is negative so the value is dropping.

12 4. The Smiths want to cover their backyard with new grass. Their backyard is rectangular, with a length of x 5 feet and a width of 4x 10 feet. However, their patio has dimensions of x + 8 by x feet. What is the area of the region of the yard that they want to cover with new grass? A. 6x 55x ft B. x + 6x 16 ft C. 11x 56x + 66 ft D. 1x 50x + 50 ft Solution: C Area to cover with new grass = area of backyard - area of patio = (4x - 10)(x - 5) - (x + 8)(x - ) = (1x - 50x + 50) - (x + 6x - 16) = 1x - 50x x - 6x + 16 = 11x - 56x Which of the following inequalities can be represented by this number line? A. 4 < b < 9 B. 9 b 4 C. b 4 or b > 9 D. b < 4 or b > 9 Solution: C 45. The graph at the right represents which equation? A. d(t) = t B. d(t) = t 1 + C. d(t) = t D. d(t) = t 1 Solution: A

13 46. Write an equation that describes the pattern shown in the table below. x y Solution: The change between each x-value is 1; the change between each y-value is - so the slope is -. So, using the first point, y - 1 = -(x - ) y = -x Simplify 5 x y Solution: x y xx y x y x 5 4 = 16 = Solve the equation 50 = 5 x + 10 Solution: Cross multiply and get 50 = 5x = 5x x = Solve the equation - ( - 4y) = y + 5 Solution: - + 4y = y y = y + 5 y = 6 y = 50. Solve the equation: x 1 < 1 Solution: -1 < x - 1 < 1 0 < x <