Inequalities Chapter Test

Transcription

1 Inequalities Chapter Test Part 1: For questions 1-9, circle the answer that best answers the question. 1. Which graph best represents the solution of 8 4x < 4 A. B. C. D. 2. Which of the following inequalities represents the graph? A. 2x y < -1 B. 2x y > -1 C. 2x y -1 D. 2x y = Find all values for x for which this inequality is true. 2/3x 4 > 10 A. x < 21 B. x < 9 C. x > 21 D. x > 9

2 4. Which of the following inequalities represents the graph: A. 1/2x +9 - ¼ < ¾ B. 1/2x +9 - ¼ > ¾ C. 1/2x +5 + ¾ >15/4 D. 1/2x +5 + ¾ < 15/4 5. Which of the following points lies in the half plane for the following inequality: 2x + 5y 10 A. (6, 4) B. (10, -1) C. (-1, 5) D. (2,1) 6. Solve. 3/4x 7 < ¼(2x 2) + 8 A. x < 58 B. x > 58 C. x < 2 D. x > 2

3 7. What is the largest integer in the solution set of the following inequality? -2x + 1/3(x-3) > -10 A. 5 B. -5 C. 6 D Lisa is renting a hall for a birthday party. They charge a $100 flat fee for the rental, plus $7 a child for food and beverages. Lisa can spend no more than $175 for the birthday party. What is the maximum number of children that can attend the birthday party on her budget? A. 9 C. 11 B. 8 D Identify the inequality that represents the following graph. A. y > x -1 B. y x-1 C. y x-1 D. y x -1

4 Part 2: For questions 10-18, complete each problem and write your answer on the blank. 10. Graph the following inequality on the graph and identify 1 solution. 3x 2y > 8 Answer: 11. Graph the following inequality on the grid. Y x-4 +1

5 12. Solve the following inequality and graph the solution on the number line. 3x 2(2x 4) < 22 Answer: 13. Solve for k: -3 < 2(k-8) +5 < Joe is selling pizza kits for a school fundraiser. Cheese pizza kits cost $12.00 and Supreme pizza kits cost $15. He would like to sell at least $200 in pizza kits. Write an inequality that represents this situation. Answer: 15. Graph the following system of inequalities on the graph and identify 1 solution. y < -3x -8 2x 2y -6 Answer:

6 16. Write an inequality that represents the following situation. Twice a number increased by 6 is at most 20. Answer: 17. Your family has no more than $40 to spend at the movies. An adult ticket cost $10 and a youth tickets costs $8. There are at least 3 youths planning on going to the movies. Write a system of inequalities to represent this situation. Answer: 18. Solve for a: 2 3/4a 6 > 6 For questions 19-21, answer each question thoroughly. (There are multiple parts to each question) 19. Tickets for the theatre production of Cinderella cost $12.00 per person. The producers were given a grant for $500 to cover the expenses of the production. Expenses for the production totaled $1200. Write an inequality that represents the number of tickets that must sold in order to make a profit. How many tickets must be sold to make a profit?

7 20. The County Soccer Club is sponsoring a fundraiser to raise money for an out of state tournament. The male soccer players are selling candy bars for $1.25 and the girl soccer players are selling flowers for $3.50. They must raise at least $500. The girls expect to sell at least 100 flowers. Write a system of inequalities to represent this situation. Graph both inequalities on the grid and shade the intersection. Give three possible solutions to this system. Justify your answer. 21. The following is a problem that Sarah solved on a math test. Analyze her response and describe whether her answer is correct or incorrect. Justify your answer mathematically. 1. 3x > 12 Math Test Sarah 3x 2 +8 > 12 3x -3+8 < -12 3x +6 > 12 3x +6 < -12 3x +6-6 > x+6-6 < x > 6 3x < -18 3x/3 > 6/3 3x/3 < -18/3 x > 2 x <-6 {-6, 2}

8 Inequalities Chapter Test Part 1: For questions 1-9, circle the answer that best answers the question. (1 point each) 1. Which graph best represents the solution of 8 4x < 4 A. B. 8-4x < 4 Original problem 8-8-4x <4-8 Subtract 8-4x < -4 Simplify (4-8 = -4) -4x/-4 <-4/-4 Divide by -4 x > 1 Reverse the sign since you divide by -4. C. D. X is greater than 1 is shown on graph B. *You can eliminate answers A and D since the inequality symbol does not contain the equal sign. < or > means the graph must have an open circle. 2. Which of the following inequalities represents the graph? A. 2x y < -1 B. 2x y > -1 C. 2x y -1 D. 2x y = -1 Notice that all of the equations are the same except for the inequality sign. We can eliminate answers C & D because the line on the graph is a dotted line. Therefore, we know that the equation must only contain a < > symbol. It cannot contain the equal sign. So, is it A or B? You must look to see which side of the line is shaded. Notice that (0,0) is not in the solution set. Therefore, we need to find the equation where (0,0) is NOT a true statement. Let s substitute to see which one works: 3. Find all values for x for which this inequality is true. A. 2(0) 0 < -1 0 < -1 is not true; therefore, this must be the correct answer. 2/3x 4 > 10 A. x < 21 B. x < 9 C. x > 21 D. x > 9 2/3x 4 > 10 Original Problem 3[2/3x -4]>10(3) Multiply by 3 to get rid of the fraction. 2x 12 > 30 Simplify 2x > Add 12 to both sides. 2x > 42 Simplify 2x/2 > 42/2 Divide by 2 on both sides. x > 21 The final answer. Letter C is the correct answer.

9 4. Which of the following equations represents the graph: A. 1/2x +9 - ¼ < ¾ B. 1/2x +9 - ¼ > ¾ C. 1/2x +5 + ¾ >15/4 D. 1/2x +5 + ¾ < 15/4 The graph shows an intersection of sets. When working with absolute value inequalities, we know that if the inequality is less than, then we have an intersection of sets. Therefore, you can automatically eliminate B and C as answers. Now we have to solve A and D to see which has the solution: -20 < x < /2x +9 - ¼ + ¼ < ¾ + ¼ Add ¼ 1/2x +9 < 1 Simplify -1 < 1/2x +9 < 1 Write as an intersection of sets -1 9 < 1/2x +9-9 < 1-9 Subtract 9 from all 3 sides -10 < 1/2x < -8 Simplify -10(2) < (2)1/2x < -8(2) Multiply by 2 to solve for x. -20 < x < -16 Letter A is the solution. 5. Which of the following points lies in the half plane for the following inequality: 2x + 5y 10 A. (6, 4) B. (10, -1) C. (-1, 5) D. (2,1) In order for the point to be a solution in the half plane, the ordered pair would need to be a true statement once it is substituted into the equation. In this case, the best method may be to guess and check! Choose an answer, substitute and check to see if it works. First, let s use some logical thinking. We know that the left hand side of the equation must be less than or equal to 10. Since we are multiplying by 2 and 5 and then adding the numbers together, I estimate that letter A will not work because the x and y coordinate are both large, positive numbers. Therefore, my answer will most likely be larger than 10. Letter B has a pretty large number for the x-coordinate and although the y-coordinate is a negative I think the answer may still be larger than 10. I can come back and try this one if the others don t work. Letter C could work because we have a negative number to substitute. So, let s try letter C. C (-1, 5) 2(-1) + 5(5) is not true, so this is not the correct answer. Letter D: Even though both coordinates are positive, they are both very small numbers, so this answer may work. Let s try. D. (2,1) 2(2) + 5(1) this is true, therefore, this is the correct answer.

10 6. Solve. 3/4x 7 < ¼(2x 2) + 8 A. x < 58 B. x > 58 3/4x 7 < ¼(2x 2) + 8 Original Problem 4 [3/4x 7] < 4[1/4(2x-2) +8 Multiply by 4 to get rid of the fraction C. x < 2 3x 28 < 2x Simplify D. x > 2 3x 28 < 2x + 30 Simplify: ( = 30) 3x -2x 28 < 2x -2x + 30 Subtract 2x from both sides x 28 < 30 Simplify ( 3x -2x = x) x < x < 58 Add 28 to both sides Final Answer - A 7. What is the largest integer in the solution set of the following inequality? -2x + 1/3(x-3) > -10 A. 5 B. -5 C. 6 D. -6-2x + 1/3(x-3) > -10 Original Problem 3 [-2x + 1/3(x-3)] > -10(3) Multiply by 3 to get rid of the fraction -6x + x-3 > -30 Simplify -5x 3 > -30 Combine like terms (-6x + x = -5x) -5x > Add 3 to both sides -5x > -27 Simplify ( = -27) -5x/-5 > -27/-5 Divide by -5 on both sides x < 27/5 or x < 5 & 2/5 Reverse the sign since you divide by -5. Since x is less than 5 & 2/5, the largest integer in the solution set is 5.

11 8. Lisa is renting a hall for a birthday party. They charge a $100 flat fee for the rental, plus $7 a child for food and beverages. Lisa can spend no more than $175 for the birthday party. What is the maximum number of children that can attend the birthday party on her budget? A. 9 B. 8 C. 11 D. 10 We first need to write an inequality to represent this situation. $100 flat fee is the y-intercept (b). $7 a child is a rate so this is the slope. Total is $no more than 175. Since we know the y-intercept and the slope, we can write this equation in slope intercept form. Y = mx + b 175 7x Since she cannot spend more than $175, 175 is the greatest amount she can spend. OR, you could reverse sides: 7x The amount that she spends must be less than or equal to 175. Now we can solve for x to find the maximum number of children than can attend the party. 7x x x 75 Original Inequality Subtract 100 from both sides Simplify 7x/7 75/7 Divide both sides by 7 X The maximum number of children that Lisa can invite is Identify the inequality that represents the following graph. A. y > x -1 B. y x-1 C. y x-1 D. y x -1 First we can eliminate letter A because the boundary line is solid, so we know that > cannot be the inequality symbol. We also know that the vertex is on the x axis, so it hasn t shifted up or down, therefore, letter D can be eliminated. The only difference between B and C is the inequality symbol. If I substitute (0,0) into the equation for B, I get: or 0 1. Since 0 is not greater than or equal to and (0,0) is not part of the shaded area, this is the correct absolute value inequality.

12 Part 2: For questions 10-18, complete each problem and write your answer on the blank. (2 points) 10. Graph the following inequality on the graph and identify 1 solution. 3x 2y > 8 I rewrote the equation in slope intercept form: 3x 3x -2y > -3x + 8 Subtract 3x from both sides -2y > -3x + 8 Simplify -2y/-2 > -3x/-2 +8/-2 Divide by -2 y < 3/2x 4 Reverse the sign since you divided by -2 When you graph this line, the line should be dotted since the inequality symbol is <. No solutions are contained on this line. Now, substitute (0,0) and shade the corresponding half plane. 3(0) 2(0) > 8 0> 8 is not true, therefore, shade the half-plane that does not contain (0,0) One solution is (10,1). Answers may vary any point in the shaded half-plane is a solution 11. Graph the following inequality on the grid. Y x Create a table of values and draw the boundary line. Choose (0,0) as your test point Since 0 is not greater than 5, do not shade the area that contains (0,0)

13 12. Solve the following inequality and graph the solution on the number line. 3x 2(2x 4) < 22 3x 2(2x 4) < 22 Original Problem 3x 4x + 8 < 22 Distribute -2 -x + 8 < 22 Simplify (3x 4x = -x) -x < 22 8 Subtract 8 from both sides -x < 14 Simplify (22-8 = 14) -x/-1 < 14/-1 Divide by -1 x > -14 Reverse your sign since you divided by Solve for k: -3 < 2(k-8) +5 < 15-3< 2k < 15 Distribute 2 throughout the parenthesis -3 < 2k -11 < 15 Simplify < 2k < Add 11 to all three sides 8 < 2k < 26 Simplify 8/2 < 2k/2 < 26/2 Divide by 2 on all three sides 4 < k < 13 Simplify

14 14. Joe is selling pizza kits for a school fundraiser. Cheese pizza kits cost $12.00 and Supreme pizza kits cost $15. He would like to sell at least $200 in pizza kits. Write an inequality that represents this situation. If we write a verbal model for this problem, I know that I would add the sales of cheese kits and supreme kits. I also know the total that I need to sell. Therefore, this equation should be written in standard form. Let c = number of cheese pizzas Let s = number of supreme pizzas Cheese + Supreme = Total Price(number) + Price (number) at least Total 12c + 15s 200 The inequality that represents this situation is: 12c + 15s Graph the following system of inequalities on the graph and identify 1 solution. y < -3x -8 & 2x 2y -6 The red line represents: The blue line represents: Y < -3x- 8 2x -2y -6 X-Intercept: (let y = 0) This equation is already 2x 2(0) = -6 written in slope intercept 2x = -6 form. Therefore, we x = -3 can plot the y-intercept (-8) and use the slope of -3 to find the next point. Y-intercept (let x = 0) 2(0) 2y = -6 The line is dotted due to -2y = -6 the < sign and no solutions y = 3 are contained on this line. We shade the left hand side Substitute (0,0) because when we substitute 2x 2y -6 (0,0): 2(0) 2(0) -6 0< -8 is not true. Therefore, 0-6 is true, therefore, we don t shade the half plane shade the half-plane that that contains (0,0). contains (0,0). 1 solution to this system of inequalities is: (-10,-10) Answers may vary. Any point located in the region that is shaded by both inequalities (purple region) is correct. 16. Write an inequality that represents the following situation. Twice a number increased by 6 is at most 20. Let x = a number Twice a # increased by 6 at most 20 2x x is the inequality that represents this situation

15 17. Your family has no more than $40 to spend at the movies. An adult ticket cost $10 and a youth tickets costs $8. There are at least 3 youths planning on going to the movies. Write a system of inequalities to represent this situation. Adult = $10 Youth = $8 No more than $40 total at least 3 youth s Let a = number of adults Let y = number of youths Since this problem asks us to write a system of inequalities, we must write 2 different inequalities. So, we must know information about two different aspects of the problem. I know information about: 1. Cost to go to the movies 2. the number of youths planning on attending. Let s write an equation to represent each of the above. Cost to go to the movies: Adults + Youths = Total Price (number) + Price (number) no more than $40 10 a + 8 y 40 10a + 8y 40 is the inequality that represents the cost to go to the movies. The number of youths attending: At least 3 youths attending y 3 (At least means greater than or equal to) The system of inequalities that represents this situation is: 10a + 8y 40 y Solve for a: 2 3/4a 6 > 6 First we must isolate the absolute value expression on the left hand side by dividing by /4a 6 > /4a 6 > 3 Divide by 2 in order to isolate the absolute value Simplify Since the inequality symbol is greater than, the solution is a union of sets and we must write two separate inequalities. 3/4a 6 > 3 or 3/4a 6 < -3 3/4a 6 >3 or 3/4a 6 < -3 3/4a 6+6 > /4a < Add 6 3/4a > 9 3/4a < 3 Simplify (4/3)3/4a > 9(4/3) (4/3)(3/4a) < 3(4/3) Multiply by 4/3 a> 12 a < 4 {a Copyright > 12 or 2009 a < 4} Algebra-class.com

16 For questions 19-21, answer each question thoroughly. (There are multiple parts to each question). (3 points each) 19. Tickets for the theatre production of Cinderella cost $12.00 per person. The producers were given a grant for $500 to cover the expenses of the production. Expenses for the production totaled $1200. Write an inequality that represents the number of tickets that must sold in order to make a profit. How many tickets must be sold to make a profit? Tickets cost $12 per person $500 towards expenses Expenses totaled $1200 Let x = # of people In order to make a profit, your sales must be greater than your expenses. Ticket sales + grant > total expenses 12x > 1200 The inequality that represents the number of tickets that must be sold in order to make a profit is: 12x > x > 1200 Original Inequality 12x > Subtract 500 from both sides 12x > 700 Simplify ( = 700) 12x/12 > 700 / 12 Divide both sides by 12 x > 58.3 They would need to sell 59 or more tickets in order to make a profit.

17 20. The County Soccer Club is sponsoring a fundraiser to raise money for an out of state tournament. The male soccer players are selling candy bars for $1.25 and the girl soccer players are selling flowers for $3.50. They must raise at least $500. The girls expect to sell at least 100 flowers. Write a system of inequalities to represent this situation. Graph both inequalities on the grid and shade the intersection. Give three possible solutions to this system. Justify your answer. Since we are asked to write a system of inequalities, we must write 2 equations. We can write 1 inequality about the sales of candy bars and flowers. We can write the 2 nd inequality about the girls selling at least 100 flowers. Let x = # of candy bars sold Let y = # of flowers sold Sales of candy bars and flowers: 1.25x y 500 Girls will sell at least 100 flowers: Y 100 The system of inequalities that represents this situation is: 1.25x y 500 y 100 In order to graph the first inequality, I must find the x and y intercepts: X intercept (let y = 0) Y-intercept (let x = 0) 1.25x + 3.5(0) = (0) +3.5y = x = y = 500 X = 400 y = Substitute (0,0) in order to determine which half plane to shade: 1.25(0) + 3.5(0) is not true. The second equation, y 100 is ready for graphing. This will be a horizontal line through (0,100) Three possible solutions to this problem would be: They could sell 200 candy bars and 120 flowers (200, 120) 1.25(200) (120) 500 y They could sell 400 candy bars and 140 flowers (400, 140) 1.25(400) (140) 500 y They could sell 600 candy bars and 180 flowers (600, 180) 1.25(600) (180) 500 y Answers will vary. Any ordered pair within the purple shaded region is correct. Substitute (0,0) in order to determine which half plane to shade: is not true.

18 21. The following is a problem that Sarah solved on a math test. Analyze her response and describe whether her answer is correct or incorrect. Justify your answer mathematically. 1. 3x > 12 Math Test Sarah 3x 2 +8 > 12 3x -3+8 < -12 3x +6 > 12 3x +6 < -12 3x +6-6 > x+6-6 < x > 6 3x < -18 3x/3 > 6/3 3x/3 < -18/3 x > 2 x <-6 {-6, 2} Sarah s answer is not correct. She made her mistake in the first line of her solution. She did not isolate the absolute value expression. The first line and the rest of the problem should be: 3x > x -2 > 4 3x 2 > 4 or 3x 2 < -4 3x -2+2 > 4+2 3x 2+2 < x > 6 3x < -2 3x/3 > 6/3 3x/3 < -2/3 x > 2 x < -2/3 Solution: {-2/3, 2} This test is worth 36 points.