2-8 two-variable - mr. gongora's website · 2015-07-07 · problem 1 got it? lesson 2-8...

114 Chapter 2 Functions, Equations, and Graphs

2-8 Two-Variable InequalitiesObjective To graph two-variable inequalities

Lesson Vocabulary

• linear inequality• boundary• half-plane• test point

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In some situations you need to compare quantities. You can use inequalities for situations that involve these relationships: less than, less than or equal to, greater than, and greater than or equal to.

Essential Understanding Graphing an inequality in two variables is similar to graphing a line. Th e graph of a linear inequality contains all points on one side of the line and may or may not include the points on the line.

A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane bounded by a line. Th is line is the boundary of the graph. Th e boundary separates the coordinate plane into two half-planes, one of which consists of solutions of the inequality.

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You have a gift card for a store that sells pre-owned CDs and paperback books. You want to spend as much of the gift card as possible. How many of each item can you buy? Explain.

The words “as muchcas possible” are important here.

Dynamic ActivityLinear Inequalities

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Content StandardsA.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Also F.IF.7.b

MATHEMATICAL PRACTICES

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Problem 1

Got It?

Lesson 2-8 Two-Variable Inequalities 115

To determine which half-plane to shade, pick a test point that is not on the boundary. Check whether that point satisfi es the inequality. If it does, shade the half-plane that includes the test point. If not, shade the other half-plane. Th e origin, (0, 0), is usually an easy test point as long as it is not on the boundary.

Graphing Linear Inequalities

What is the graph of each inequality?

A y S 3x 2 1

Step 1 Step 2 Graph the boundary line y 5 3x 2 1. Choose a test point, (0, 0). Substitute x 5 0

Use a dashed boundary line because the and y 5 0 into y . 3x 2 1.inequality is greater than, and the points 0 . 3(0) 2 1on the line do not satisfy the inequality. 0 . 21

Since 0 . 21 is true, shade the half plane that includes (0, 0).

B y K 3x 2 1

Th e boundary line is again y 5 3x 2 1, but it is solid because the inequality is less than or equal to.

Shade the region opposite the region shaded above (for .) because the inequality is less than or equal to.

You can also check the point (0, 0).

0 # 3(0) 2 1

0 # 21

Since 0 # 21 is false, (0, 0) is not part of the solution.

1. What is the graph of each inequality? a. y $ 22x 1 1 b. y , 22x 1 1

You can also inspect inequalities solved for y, such as y . mx 1 b to determine which half-plane describes the solution. Since y describes vertical position, the solution of y . mx 1 b will be above the boundary line. Th e solution of y , mx 1 b will be below the boundary line.

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Can you use the graph of y S 3x 2 1 to help graph y K 3x 2 1?If you shaded above the line for y . 3x 2 1, then shade below the line for y # 3x 2 1.


Problem 2

Got It?


Using a Linear Inequality

Entertainment Th e map shows the number of tickets needed for small or large rides at the fair. You do not want to spend more than $15 on tickets. How many small or large rides can you ride?

You can buy 60 tickets with $15.

Relate the number of

tickets for small rides plus the number of

tickets for large rides is less than or equal to 60

Defi ne Let x 5 the number of small rides.

Let y 5 the number of large rides.

Write 3x 1 5y # 60

Step 1

Find the intercepts of the boundary line. Use the intercepts to graph the boundary line.

When y 5 0, 3x 1 5(0) 5 60. When x 5 0, 3(0) 1 5y 5 60.

3x 5 60 5y 5 60

x 5 20 y 5 12

Graph the line that connects the intercepts (20, 0) and (0, 12). Since the inequality is #, use a solid boundary line.

Step 2

Th e region above the boundary line represents combinations of rides that require more than 60 tickets.You purchased a fi nite number of tickets, 60, so you will not be able to go on an infi nite number of rides. Shade the region below the boundary line.

Th e number of small rides x and the number of large rides y are whole numbers. In math, such a situation is called discrete. All points with whole number coordinates in the shaded region represent possible combinations of small and large rides.

2. a. Suppose that you decide to spend no more than $30 for tickets. What are the possible combinations of small and large rides that you can ride now? Use a graph to fi nd your answer.

b. Reasoning Why did the graph of the solution in Problem 2 only include Quadrant I?

What are the unknowns?The unknowns are the number of small rides and the number of large rides you can get on.

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Problem 3

Problem 4

Got It?

Got It?


You can graph two-variable absolute value inequalities in the same way that you graph linear inequalities.

Graphing an Absolute Value Inequality

What is the graph of 1 2 y R »x 1 2…?

1 2 y , u x 1 2 u

2y , u x 1 2 u 2 1 Subtract 1 from each side.

y . 2u x 1 2 u 1 1 Multiply both sides by 21.

Th e graph of y 5 2u x 1 2 u 1 1 is the graph of y 5 u x u , refl ected in the x-axis and translated left 2 units and up 1 unit.

Since the inequality is solved for y and y . 2u x 1 2 u 1 1, shade the region above the boundary.

3. What is the graph of y 2 4 $ 2 u x 2 1 u ?

You can use the transformations discussed in previous lessons to help draw the boundary graphs more quickly. You can also use them to write an inequality based on a graph.

Writing an Inequality Based on a Graph

What inequality does this graph represent?

Th e boundary is the graph of the absolute value function y 5 u x u , translated. Th e vertex of y 5 u x u is translated to (3, 22), so the boundary is the graph of y 5 u x 2 3 u 2 2.

Th e solution is shaded above the boundary, so the inequality is either . or $. Since the boundary is a dashed line, the correct inequality is y . u x 2 3 u 2 2.

4. a. What inequality does this graph represent? b. Reasoning You can tell from looking at the inequality

y . 5x 2 3 to shade above the boundary line to represent the solution. Can you use the same technique to show the solution of an inequality like 2x 2 y . 1? Explain.

• Solve the inequality for y.• Graph the related equation. • Shade the solution.

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BoundaryBAbsolute value inequality

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How can you tell that the graph is not a stretch or compression of the graph of y 5 »x…?The slopes of the branches are 1 and 21.



Do you know HOW?What is the graph of each inequality?

1. 9y # 12x

2. 7x 1 y $ 8

What is the graph of each absolute value inequality?

3. y # u x 1 1 u

4. y $ u 2x 2 3 u

Do you UNDERSTAND? 5. Do the points on the boundary line of the graph of

an inequality help determine the shaded area of the graph? Explain.

6. Compare and Contrast How is graphing a linear inequality in two variables diff erent from graphing a linear equation in two variables?

7. Reasoning Is the ordered pair Q34, 0R a solution of

3x 1 y . 3? Explain.

Lesson Check

Practice and Problem-Solving Exercises

Graph each inequality.

8. y . 2x 1 1 9. y , 3 10. x # 0

11. y # x 2 5 12. 2x 1 3y $ 12 13. 2y $ 4x 2 6

14. 3x 2 2y # 9 15. 2y , 2x 1 2 16. 5 2 y $ x

17. Cooking Th e time needed to roast a chicken depends on its weight. Allow at least 20 min/lb for a chicken weighing as much as 6 lb. Allow at least 15 min/lb

for a chicken weighing more than 6 lb. a. Write two inequalities to represent the time needed to roast a chicken. b. Graph the inequalities.

Graph each absolute value inequality.

18. y $ u 2x 2 1 u 19. y # u 3x u 1 1 20. y # u 4 2 x u

21. y . u2x 1 4 u 1 1 22. y 2 7 . u x 1 2 u 23. y 1 2 # u 12 x u 24. 3 2 y $ 2u x 2 4 u 25. 1 2 y , u 2x 2 3 u 26. y 1 3 # u 3x u 2 1

Write an inequality for each graph. Th e equation for the boundary line is given.

27. y 5 2x 2 2 28. 5x 1 3y 5 9 29. 2y 5 u 2x 1 6 u

PracticeA See Problem 1.

See Problem 2.

See Problem 3.

See Problem 4.

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Graph each inequality on a coordinate plane.

30. 5x 2 2y $ 210 31. 2x 2 5y , 210 32. 34 x 1 2

3 y . 52 33. 3(x 2 2) 1 2y # 6

34. u x 2 1 u . y 1 7 35. y 2 u 2x u # 21 36. 23 x 1 2 # 2

9 y 37. 0.25y 2 1.5x $ 24

38. Think About a Plan Th e graph at the right relates the number of hours you spend on the phone to the number of hours you spend studying per week. Describe the domain for this situation. Write an inequality for the graph.

• What is the least amount of time you can spend on the phone per week? What is the most?

• What is the least amount of time you can spend studying per week? What is the most?

• What is the greatest amount of time you can spend either on the phone or studying per week?

Write an inequality for each graph.

39. 40. 41.

42. 43. 44.

45. Which graph best represents the solution of the inequality y $ 2 u x 2 1 u 2 2?

46. Th e graph at the right relates the amount of gas in the tank of your car to the distance you can drive.

a. Describe the domain for this situation. b. Why does the graph stop? c. Why is only the fi rst quadrant shown? d. Reasoning Would every point in the solution region be a

solution? e. Write an inequality for the graph. f. What does the coeffi cient of x represent?

ApplyB

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47. Writing When you graph an inequality, you can often use the point (0, 0) to test which side of the boundary line to shade. Describe a situation in which you could not use (0, 0) as a test point.

Graphing Calculator Graph each inequality on a graphing calculator. Th en sketch the graph.

48. y # u x 1 1 u 2 u x 2 1 u 49. y . u x u 1 u x 1 3 u

50. y , u x 2 3 u 2 u x 1 3 u 51. y , 7 2 u x 2 4 u 1 u x u

ChallengeC

Mixed Review

Graph each function by translating its parent function.

56. y 5 u 2x 1 5 u 57. y 5 u x u 2 3 58. f (x) 5 u x 1 6 u

59. f (x) 5 u x u 2 2 60. y 5 u x 1 2 u 61. y 5 u x 2 1 u 1 5

Determine whether y varies directly with x. If so, fi nd the constant of variation.

62. y 5 x 1 1 63. y 5 100x 64. 5x 1 y 5 0 65. y 2 2 5 2x

66. x 5y3 67. 24 5 y 2 x 68. y 5 210x 69. xy 5 1

Make a scatter plot and describe the correlation.

70. 5(0, 6), (1, 4), (2, 4), (4, 1), (5, 0)6 71. 5(210, 5), (25, 25), (22, 0), (0, 3), (5, 22)6

Get Ready! To prepare for Lesson 3-1, do Exercises 72–74.

Graph each equation. Use one coordinate plane for all three graphs.

72. 3x 2 y 5 2 73. 3x 2 y 5 22 74. x 1 3y 5 22

See Lesson 2-7.

See Lesson 2-2.

See Lesson 2-5.

See Lesson 2-3.

Standardized Test Prep

52. Suppose y varies directly with x. If x is 30 when y is 10, what is x when y is 9?

3 27 29 3009

53. Which equation represents a line with slope 22 and y-intercept 3?

3y 5 x 2 2 3y 5 22x 1 1 y 5 2x 2 3 y 5 22x 1 3

54. What is the vertex of y 5 u x u 2 5?

(5, 0) (25, 0) (0, 5) (0, 25)

55. Th e amount of a commission is directly proportional to the amount of a sale. A realtor received a commission of $48,000 on the sale of an $800,000 house. How much would the commission be on a $650,000 house?

SAT/ACT

ExtendedResponse


2-8 two-variable - mr. gongora's website · 2015-07-07 · problem 1 got it? lesson 2-8...

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