Solving and Graphing Linear Inequalities Chapter Questions 1. How do we translate a statement into an inequality? 2. Explain the steps to graphing an inequality on a number line. 3. How is solving an inequality much like solving a linear equation? 4. What does the and & or tell us in a compound inequality? 5. When do the special cases happen with compound inequalities? 6. How do graphs of linear inequalities compare with graphs of equations? 7. How do you determine what type of line to use? 8. How do you determine where to shade? NJ Center for Teaching and Learning ~ 1 ~ www.njctl.org
Solving Linear Inequalities Chapter Problems Simple Inequalities Involving Addition and Subtraction Classwork 1. Solve, check and graph the following inequalities. a. x + 5 > 10 b. x + 3 < -2 c. 7 > x + 11 d. x 3-5 e. x 7 > 3 f. -2 + x -2 g. -9 x 3 h. x + 0.5 < 4 i. -3.75 x 1.25 j. 8.7 > x + 2.2 2. Write an inequality for each sentence below and then solve and check it. a. The sum of w and nine is less than 18. b. g decreased by 25 is at most five. c. The difference of a number and six is no less than 15. d. 14 is more than the sum of ten and a number. e. 25 plus a number is at least 13. 3. Suppose you must maintain at least $500 in your checking account in order to have free checking. Your balance is $542 and then you write a check for $57. How much do you need to deposit in order to keep your free checking? Write an inequality and solve. 4. You need no more than 2,200 calories in a day. You had 650 calories at breakfast and 825 calories at lunch. At most, how many calories, c can you eat for dinner? Write an inequality and solve. Homework 5. Solve, check and graph the following inequalities. a. x + 7 > -2 b. x + 3 < -3 c. -8 < x + 15 d. x 4 1 e. x 1 > 6 f. -7 + x -11 g. -6 x 2 h. x + 2.5 < 6 NJ Center for Teaching and Learning ~ 2 ~ www.njctl.org
i. -5.5 x 3.25 j. 7.9 > x + 4.4 6. Write an inequality for each sentence below and then solve and check it. a. The difference of a number and seven is at most 16. b. 18 is less than a number plus 7 c. h decreased by 3 is more than 1. d. 14 is greater than or equal to the sum of 18 and a number. e. The sum of b and 22 is at least 6 7. Suppose you must maintain at least $500 in your checking account in order to have free checking. Your balance is $612 and then you make a deposit of $79. How much can you withdraw and still keep your free checking? Write an inequality and solve. 8. You need no more than 2,200 calories in a day. You had 720 calories at breakfast and plan on having 1,000 calories at dinner. How many calories, c can you eat for lunch? Write an inequality and solve. Simple Inequalities Involving Multiplication and Division Classwork 9. Solve, check and graph the following inequalities. a. 5x > -25 b. -7x -21 c. 18 > 2x d. 25x 100 e. -30-6x f. 10x < 0 g. 8x 24 h. 40 < -8x i. 20x 30 j. 350 > -70x 10. Solve, check and graph the following inequalities.! a. " 2 b.! # < 14 c. -3! $% d.! $& > 1 NJ Center for Teaching and Learning ~ 3 ~ www.njctl.org
e. f.! $' -3! ( 3 g. 0! ) h. -1! #." i. j.! $+ < 2.2! $+." > -10 11. Write an inequality for each sentence below and then solve and check it. a. The product of r and 5 is no more than 55. b. The quotient of v divided by -4 is greater than or equal to 2. c. Half of d is greater than 40. d. Twice a number is at most 24. e. One-fourth of y is less than or equal to -12. f. The product of -8 and x is no less than -64. 12. What happens to the inequality symbol when you do each of the following to both sides of an inequality? a. Multiply by a positive number b. Divide by a positive number c. Add a negative number d. Subtract a negative number e. Divide by a negative number f. Multiply by a negative number Homework 13. Solve, check and graph the following inequalities. a. 4x -16 b. 32 8x c. -7x > 49 d. -5x < -55 e. 13x 0 f. 60 < 12x g. 66 > -3x h. -6x 360 NJ Center for Teaching and Learning ~ 4 ~ www.njctl.org
i. -2x 17 j. 26 < 4x 14. Solve, check and graph the following inequalities. a. 5! ( b. c. d.! ( < -3! $) 0! ( > 11 e. 4! $+ f. -4! $% g. h.! $' -1.5! ' 12 i. j.! $% > -9! $# < 3.3 15. Write an inequality for each sentence below and then solve and check it. a. Four times a number n is no more than 24. b. One-third of m is greater than 9. c. The product of -9 and g is at most 81. d. Half of d is less than or equal to 16. e. The quotient of v divided by -6 is less than 4. f. The product of 7 and x is no less than -42. NJ Center for Teaching and Learning ~ 5 ~ www.njctl.org
Two-Step and Multi-Step Inequalities Classwork 16. Solve, check and graph the following inequalities. a. 10 + 7x < 17 b. -1 + 3x -28 c. -4x + 2 > -46 d. 3 + 5x + 3x 19 e. -6x 6 + 3x > -24 f. -4 2x + 5x 8 g. 6x 8 > 1 + 3x h. 3x 9 > -41 + 7x i. 4(-7 + 3x) 32 j. 5(-7x + 8) 275 k. ( ' x - + # + ' l. x + %. < ( #) 17. Write an inequality for each situation, then solve. a. You want to spend $20 for a taxi ride. Before you go anywhere, the taxi driver sets the meter the initial charge of $2. The meter adds $1.50 for each mile and you plan on giving a $2 tip, what is the farthest you can go? b. On a trip from New Jersey to Florida, your family wants to travel at least 436 miles in one day from 8am to 5 pm. You plan on only stopping for a total of 1 hour. What must your average speed be? Homework 18. Solve, check and graph the following inequalities. a. 5x + 1 > 51 b. 6x + 6 24 NJ Center for Teaching and Learning ~ 6 ~ www.njctl.org
c. -6x 3 > -63 d. 4 3x 5x 36 e. 3 + x + 6x < -53 f. 2 + 2x 3x + 2 g. -8 + 7x 4x -52 h. 2x + 8 > -x -1 i. 4(-3x -5) < -164 j. 6(3x + 9) 126 k. # x + ' > (+ (. '# l. - + ' - # ( x " '' 19. Write an inequality for each situation, then solve. a. You have $50 to spend at the boardwalk. Tickets for the rides cost $1.50 and parking is $9. What is the most number of tickets you can buy? b. You are shopping at the mall and have less than $75 to spend. If you bought 3 shirts and spent $18 on snacks, how much could one shirt be, is all were the same price? Compound Inequalities & Special Cases Classwork 20. Write the compound inequality for the graph below. 21. Write the compound inequality for the graph below. 22. Write the compound inequality for the graph below. NJ Center for Teaching and Learning ~ 7 ~ www.njctl.org
23. Write the compound inequality for the graph below. 24. Solve each inequality and graph the solution on a number line. a. 7 x + 4 < 10 b. 4 < x + 6< -2 c. 15 x 3 9 d. 3x > -9 and 8 < 2x e. x + 3 > 5 and -4x -8 f. 3x < 5 and 2x 2-3 g. -3x > -12 and 2x > 6 h. x + 3 > 15 or x 7 < -3 i. 4x > -12 or 2x > 12 j. 2x + 4 > 12 or 3x + 12 < 15 k. 5x + 7 < 12 or -7x > -21 l. 4x 20 or 3x + 1 > -5 25. Model each situation with an inequality and solve. a. Your friend is on a diet. He is supposed to eat at least 1500 but no more than 1800 calories per day. Before dinner he has had 1050 calories. What number of calories should he eat at dinner? b. To get an A in class your total points must be between 690 and 750, inclusive. If you have 652 points before the last test. What possible numbers of points could you earn and still get an A in the class. Compound Inequalities & Special Cases Homework 26. Write the compound inequality for the graph below. 27. Write the compound inequality for the graph below. NJ Center for Teaching and Learning ~ 8 ~ www.njctl.org
28. Write the compound inequality for the graph below. 29. Write the compound inequality for the graph below. 30. Solve each inequality and graph the solution on a number line. a. -4 < x + 2 < 9 b. 7 3 2x < 11 c. 4 x + 3 < 9 d. x 0 and x < 25 e. 8x > -64 and 8 > 4x f. -2 > 3x + 7 and 3x + 1-14 g. x + 3 < 12 and 4x > 24 h. -9x 3 < 6 or 3x + 5 26 i. 6x 8 > 10 or -9x > -9 j. 10x -20 or 20x 40 k. -4x + 3 > -17 or 4x 8 4 l. -7x + 10 < 31 or 10x + 20-50 31. Model each situation with an inequality and solve. a. The sum of any two sides of a triangle is greater than the length of the third side. If two sides of a triangle are 5 inches and 17 inches. Find the range of values for the lengths of the third side. b. A gymnast wants her average score after 3 events to be between 9.0 and 10. If she scores an 8.7 and 9.1, what possible values for the third event will make her average between 9.0 and 10? Classwork Graph each linear inequality. 32. 2y 4x < -2 33. y < 10x + 4 34. y > 8x 35. y > -½x 4 36. -3y < -9x + 3 Graphing Linear Inequalities NJ Center for Teaching and Learning ~ 9 ~ www.njctl.org
37. 6x 4y < -12 38. 16x + 6y > 36 39. y > 5 6 x 8 Write the inequality described then graph the inequality. 40. Suppose your class is raising money for their school trip. You make $5 on each smoothie and $3 on each soft pretzel that you sell. How many items of each type must you sell to raise more than $150? a. Write a linear inequality that describes the situation. b. Graph the linear inequality. c. Write two possible solutions to the problem. Homework Graph each linear inequality. 41. y > -4 42. x + 4y < 8 43. -5x + 4y > -24 44. y < 5 3 x + 5 45. 2x + 3y < -15 46. y > -2x + 7 47. y < 9 4 x 6 48. x + 3y < 12 Write the inequality described then graph the inequality. 49. Suppose that you exercise by either running 5 miles per day or rollerblading 10 miles per day. How many days will it take you to cover a distance of at least 150 miles? a. Write a linear inequality that describes the situation. b. Graph the linear inequality. c. Write two possible solutions to the problem. NJ Center for Teaching and Learning ~ 10 ~ www.njctl.org
Solving and Graphing Linear Inequalities Unit Review Multiple Choice Choose the correct answer for each question. No partial credit will be given. 1. Which value of x is in the solution set of the inequality 7(x + 1) > -14? a. -22 b. -2 c. -1 d. 1 2. Which graph represents the solution set for: 1 / 5 + 1 / 3 x 4 / 15 a. b. c. d. 3. Which type of line is used when graphing -4x - 3y 10 a. Dashed b. Solid c. No line For #4 and #5, Determine which point is a solution to each linear inequality. 4. 4x - 2y 6 a. (7, 9) b. (0, -4) c. (-1, -8) d. (-10, -100) NJ Center for Teaching and Learning ~ 11 ~ www.njctl.org
5. -5x - 3y > 10 a. (-2, 0) b. (0, -2) c. (-3, -1) d. (1, -4 ) For #6 and #7, Choose the linear inequality that describes each graph. 6. a. 4y < -3x +12 b. 3x 4y > 12 c. y > ¾ x + 3 d. 3x + 4y > 12 7. a. y - 10x + 11 b. 10x + 2y 11 c. y - 5x + 5.5 d. 10x 2y 11 Short Constructed Response Write the correct answer for each question. No partial credit will be given. 8. Solve and graph: 3x +2 < 0 NJ Center for Teaching and Learning ~ 12 ~ www.njctl.org
9. Solve and Graph: ¼ x 3 / 5 2 / 3 10. Solve and Graph: 5x 4 6 and 7x + 11 < 32 11. Solve and Graph: 22 2(2m 1) or 5 3m -13 12. Graph: 7x + 4 < 18 13. Graph: 3x + y > 1 14. Graph: y < - 1 / 9 x - 1 15. Write a linear inequality with (0, 4) as a solution. 16. Write a linear inequality with (4, -2) on the boundary but not a solution. NJ Center for Teaching and Learning ~ 13 ~ www.njctl.org
17. Write the inequality for the graph below. 18. Write the inequality for the graph below. 19. Write the inequality for the graph below. 20. Most snakes live where the temperatures ranges from at least 75 F to no more than 90 F. Graph the temperatures where snakes will not thrive. Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given. 21. Model with an inequality and solve. Graph your solution. Maria is in charge of organizing the school s holiday carnival. She is setting up booths in the gym. Each booth can have three players. Maria plans on setting up at least 22 booths in the gymnasium. What is the minimum number of players that can compete in the booths at the holiday carnival? NJ Center for Teaching and Learning ~ 14 ~ www.njctl.org
22. The fast food restaurant in town is selling hot dogs and hamburgers to raise money for a local charity. They are selling the hot dogs for $2.00 each and the hamburgers for $3.50 each. What is the minimum number of hot dogs and hamburgers the restaurant can sell in order to raise at least $1000 for the charity? a. Write an inequality that describes the situation. b. Graph the inequality. c. Write three possible solutions to the problem. d. Why is 200 hot dogs and 200 hamburgers not a solution? NJ Center for Teaching and Learning ~ 15 ~ www.njctl.org
1. a. x>5 Answer Key b. x<-5 c. -4 x d. x<-2 e. x>10 f. x 0 g. -6<x h. x<3.5 i. -2.5<x j. 6.5>x 2. a. w+9<18 w<9 b. g-25<5 g<30 c. x-6 15 x 21 d. 14 10+x 4 x e. 25+x 13 x -12 3. 542-57+x 500 x 15 4. 650+825+c<2200 c<725 5. a. x>-9 b. x<-6 NJ Center for Teaching and Learning ~ 16 ~ www.njctl.org
c. -23<x d. x<5 e. x>7 f. x -4 g. -4<x h. x<3.5 i. -2.25<x j. 3.5>x 6. a. x-7<16 x<23 b. 18<x+7 11<x c. h-3>1 h>4 d. 14 18+x -4 x e. b+22 6 b -16 7. 612+79-x 500 191 x 8. 720+1000+c<2200 c<580 9. a. x>-5 b. x 3 c. 9>x NJ Center for Teaching and Learning ~ 17 ~ www.njctl.org
d. x 4 e. 5 x f. x<0 g. x 3 h. 5 x i. x 3/2 j. 5<x 10. a. x 10 b. x<28 c. 18 x d. x<-9 e. x<12 f. x<9 g. 0<x h. -2.5 x i. x>-2.2 NJ Center for Teaching and Learning ~ 18 ~ www.njctl.org
j. x<15 11. 12. a. 5r<55 r<11 b. v/(-4) 2 v<-8 c. (d/2)>40 d>80 d. 2x<24 x<12 e. (y/4) <-12 y <48 f. -8x -64 x<8 a. Stays the same b. Stays the same c. Stays the same d. Stays the same e. Change to the opposite inequality sign f. Change to the opposite inequality sign 13. a. x<-4 b. 4<x c. x<-7 d. x>11 e. x 0 f. 5<x g. -22<x h. x<-60 i. x<-8.5 NJ Center for Teaching and Learning ~ 19 ~ www.njctl.org
j. 6.5<x 14. a. 15<x b. x<-9 c. x<0 d. x>33 e. -4 x f. 24<x g. x<6 h. x<48 i. x<54 j. x>-6.6 15. a. 4n<24 n<6 b. (m/3)>9 m>27 c. -9g<81 g -9 d. (d/2) <16 d<32 e. (v/-6)<4 v>-24 f. 7x -42 x -6 16. a. x<1 NJ Center for Teaching and Learning ~ 20 ~ www.njctl.org
b. x -9 c. x<12 d. x<2 e. x<8 f. x<4 g. x>3 h. 8>x i. x 5 j. x -47/7 k. x<1/3 l. x>.75 17. a. 2+1.5x+2<20 x<32/3 b. 9x-1x 436 x 54.5 18. a. x>10 b. x<3 NJ Center for Teaching and Learning ~ 21 ~ www.njctl.org
c. x<10 d. x<4 e. x<-8 f. 0<x g. x<-44/3 h. x>-3 i. x>12 j. x 4 k. x>.25 l. x -6/11 19. a. 27 b. x<19 20. -5<x<5 21. -7<x<3 22. x<-4 and 0<x 23. x<1 and x>3 24. a. 3<x<6 b. x -2 and x<-8 c. x<12 and x 18 NJ Center for Teaching and Learning ~ 22 ~ www.njctl.org
d. x>-3 and 4<x e. x>2 and x<2 f. x< (5/3) and x<-.5 g. x<4 and x 3 h. x>12 or x<4 i. x -3 or x>6 j. x 4 or x<1 k. x<1 or x 3 l. x<5 or x>2 25. a. 450<x<750 b. 38<x<398 26. -4<x<6 27. -9<x<-3 28. x<-2 and 5<x 29. x<-6 and -2<x 30. a. -6<x<7 b. -2 x>-4 c. 1<x<6 NJ Center for Teaching and Learning ~ 23 ~ www.njctl.org
d. x 0 and x<25 e. x 8 and 2>x f. -3>x and x< -5 g. x<9 and x 6 h. x>-1 or x 7 i. x -3 or x<1 j. x< -2 or x 2 k. x< 6 or x 3 l. x -3 or x< -7 31. a. 12<x<22 b. 9.2 < x<12.2 CLASSWORK 32. 2y 4x - 2 +4x +4x 2y 4x 2 y 2x 1 (Solid boundary line) test (0,0) 0 2(0) 1 0-1 (FALSE, shade region NOT containing the test point) NJ Center for Teaching and Learning ~ 24 ~ www.njctl.org
33. y < 10x +4 test (0,0) (Dashed boundary line) 0 < 10(0) + 4 0 < 4 (TRUE, shade region containing the test point) 34. y 8x (Solid Boundary line) test (1, 0) 0 8(1) 0 8 (FALSE, shade region NOT containing the test point) 35. y > - 1/2x - 4 test (0, 0) (Dashed boundary line) 0 > - 1/2(0) - 4 0 > - 4 (TRUE, shade region containing the test point) NJ Center for Teaching and Learning ~ 25 ~ www.njctl.org
36. - 3y < - 9x + 3 y > 3x 1 (Dashed boundary line) test (0, 0) 0 > 3(0) - 1 0 > - 1 (TRUE, shade region containing the test point) 37. 6x 4y < - 12 6x < 4y 12 6x +12 < 4y 3/2x +3 < y y > 3/2x +3 (Dashed boundary Line) test (0, 0) 0 > 3/2(0) + 3 0 > 3 (FALSE, shade region NOT containing the test point) 38. 16x + 6y > 36-16x - 16x 6y > - 16x + 36 y > - 8/3x + 6 (Dashed boundary line) test (0, 0) 0 > - 8/3(0) + 6 0 > 6 (FALSE, shade region NOT containing the test point) NJ Center for Teaching and Learning ~ 26 ~ www.njctl.org
39. y 6/5x - 8 (Solid boundary line) test (0, 0) 0 6/5(0) - 8 0-8 (TRUE, shade region containing the test point) 40. a. 5x+3y>150, (x: smoothie, y: soft pretzel) b. (Dashed boundary line) Using the intercepts method: set x=0, y=50 - - y- intercept (0, 50) set y=0, x=30 x- intercept (30, 0) test (0, 0) 5(0) +3(0) > 150 0 > 150 (FALSE, shade region NOT containing the test point) c. Multiple Answers ex: (25, 25) (Or, choose any point with coordinates in the shaded region) NJ Center for Teaching and Learning ~ 27 ~ www.njctl.org
HOMEWORK 41. y - 4 (Solid boundary line) test (0, 0) 0-4 (TRUE, shade region containing the test point) 42. x + 4y < 8 - x - x 4y < - x + 8 y < - 1/4x + 2 (Dashed Boundary Line) test (0, 0) 0 < - 1/4(0) + 2 0 < 2 (TRUE, shade region containing the test point) 43. - 5x + 4 y - 24 +5x +5x 4y 5x 24 y 5/4x 6 (Solid Boundary Line) test (0, 0) 0 5/4(0) 6 0-6 (TRUE, shade region containing the test point) NJ Center for Teaching and Learning ~ 28 ~ www.njctl.org
44. y 3/5x +5 (Solid Boundary Line) test (0, 0) 0 3/5(0) +5 0 5 (TRUE, shade region containing the test point) 45. 2x + 3y < - 15-2x - 2x 3y < - 2x 15 y < - 2/3x 5 (Dashed Boundary Line) test (0, 0) 0 < - 2/3(0) 5 0 < - 5 (FALSE, shade region NOT containing the test point) 46. y > - 2x +7 (Dashed Boundary line) test (0, 0) 0 > - 2(0) + 7 0 > 7 (FALSE, shade region NOT containing the test point) NJ Center for Teaching and Learning ~ 29 ~ www.njctl.org
47. y 4/9x 6 ( Solid Boundary line) test (0, 0) 0 4/9(0) 6 0-6 (FALSE, shade region NOT containing the test point) 48. - x + 3y < - 12 +x +x 3y < x 12 y < 1/3x 4 (Dashed Boundary line) test (0, 0) 0 < 1/3(0) - 4 0 < - 4 (FALSE, shade region NOT containing the test point) 49. a. 5x+10y 150 (x: running, y: rollerblading) b. (Solid boundary line) Using the intercepts method: set x=0, y=15 - - y- intercept (0, 15) set y=0, x=30 x- intercept (30, 0) test (0, 0) 5(0) +10(0) 150 0 150 (FALSE, shade region NOT containing the test point) c. Multiple Answers ex: (12,15) (or, choose any point with coordinates in the shaded region) NJ Center for Teaching and Learning ~ 30 ~ www.njctl.org
Solving and Graphing Linear Inequalities Review Answer key 1.) d 2.) b 3.) b 4. a 5. c 6. d 7. b 8. 3x + 2 < 0 9. ¼ x 3 / 5 2 / 3 10. 5x 4 6 and 7x + 11 < 32 11. 22 2(2m 1) or 5 3m -13 NJ Center for Teaching and Learning ~ 31 ~ www.njctl.org
12. 7x + 4 < 18 13. 3x + y > 1 14. y < - 1 / 9 x - 1 15. Student solutions will vary. Sample Answers: 2x + 3 y x + y 1 NJ Center for Teaching and Learning ~ 32 ~ www.njctl.org
16. Student solutions will vary. Sample answers: 9x 38 > -2 x + y > 2 17. Student solutions will vary. Sample answers: 2x + 6 < 9 x < 1.5 18. Student solutions will vary. Sample answers: -5x 1 20 x -4.2 19. Student solutions will vary. Sample answer: 3x + 1 > 4 and 7x 2 < 12 20. 21. 22.) a. 2x + 3.50y 1000 c. Possible solutions: (400, 200) (200, 400) (0, 300) d. By only selling 200 hot dogs and 200 hamburgers, the restaurant will not make enough to meet their goal of $1000. NJ Center for Teaching and Learning ~ 33 ~ www.njctl.org