2.8 Graph Linear Inequalities in Two Variablesboisemath.weebly.com/uploads/2/6/4/3/26433359/chap._2.8_notes.pdf · 2.8 Graph Linear Inequalities in Two Variables 133 KEY CONCEPT For

132 Chapter 2 Linear Equations and Functions

Key Vocabulary • linear inequality in

two variables• solution of a linear

inequality• graph of a linear

inequality• half-plane

Before You solved linear inequalities in one variable.

Now You will graph linear inequalities in two variables.

Why? So you can model data encoding, as in Example 4.

2.8 Graph Linear Inequalities in Two Variables

A linear inequality in two variables can be written in one of these forms:

Ax 1 By < C Ax 1 By ≤ C Ax 1 By > C Ax 1 By ≥ C

An ordered pair (x, y) is a solution of a linear inequality in two variables if the inequality is true when the values of x and y are substituted into the inequality.

✓ GUIDED PRACTICE for Example 1

Tell whether the given ordered pair is a solution of 5x 2 2y ≤ 6.

1. (0, 24) 2. (2, 2) 3. (23, 8) 4. (21, 27)

GRAPHING INEQUALITIES The graph of a linear inequality in two variables is the set of all points in a coordinate plane that represent solutions of the inequality.

x

y

1

1

3x 2 2y > 2

The boundary line divides the plane into two half-planes. The shaded half-plane is the graph of 3x 2 2y > 2.

All solutions of 3x 2 2y > 2 lie on one side of the boundary line 3x 2 2y 5 2.

INTERPRET GRAPHS

A dashed boundary line means that points on the line are not solutions. A solid boundary line means that points on the line are solutions.

★

Solution

Ordered Pair Substitute Conclusion

(6, 23) 3(6) 1 4(23) 5 6 >/ 8 (6, 23) is not a solution.

(0, 2) 3(0) 1 4(2) 5 8 >/ 8 (0, 2) is not a solution.

(22, 21) 3(22) 1 4(21) 5 210 >/ 8 (22, 21) is not a solution.

(23, 5) 3(23) 1 4(5) 5 11 > 8 (23, 5) is a solution.

c The correct answer is D. A B C D

E X A M P L E 1 Standardized Test Practice

Which ordered pair is a solution of 3x 1 4y > 8?

A (6, 23) B (0, 2) C (22, 21) D (23, 5)

n2pe-0208.indd 132n2pe-0208.indd 132 10/20/05 10:17:13 AM10/20/05 10:17:13 AM

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2.8 Graph Linear Inequalities in Two Variables 133

KEY CONCEPT For Your Notebook

Graphing a Linear Inequality

To graph a linear inequality in two variables, follow these steps:

STEP 1 Graph the boundary line for the inequality. Use a dashed line for < or > and a solid line for ≤ or ≥.

STEP 2 Test a point not on the boundary line to determine whether it is a solution of the inequality. If it is a solution, shade the half-plane containing the point. If it is not a solution, shade the other half-plane.

Graph (a) y ≤ 23 and (b) x < 2 in a coordinate plane.

a. Graph the boundary line y 5 23. b. Graph the boundary line x 5 2. Use a solid line because the Use a dashed line because the inequality symbol is ≤. inequality symbol is <.

Test the point (0, 0). Because Test the point (0, 0). Because (0, 0) is not a solution of the (0, 0) is a solution of the inequality, shade the half-plane inequality, shade the half-plane that does not contain (0, 0). that contains (0, 0).

E X A M P L E 2 Graph linear inequalities with one variable

x

y

3(0, 0)

y ≤ 23

21

x

y

3

1

(0, 0)x < 2

Graph (a) y > 22x and (b) 5x 2 2y ≤ 24 in a coordinate plane.

a. Graph the boundary line y 5 22x. b. Graph the boundary line Use a dashed line because the 5x 2 2y 5 24. Use a solid line inequality symbol is >. because the inequality symbol is ≤.

Test the point (1, 1). Because Test the point (0, 0). Because(1, 1) is a solution of the (0, 0) is not a solution of theinequality, shade the half-plane inequality, shade the half-planethat contains (1, 1). that does not contain (0, 0).

E X A M P L E 3 Graph linear inequalities with two variables

x

y

1

2

(1, 1)

y > 2 2xx

y

3(0, 0)

5x 2 2y ≤ 24

23

at classzone.com

AVOID ERRORS

It is often convenient to use (0, 0) as a test point. However, if (0, 0) lies on the boundary line, you must choose a different test point.


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E X A M P L E 4 Solve a multi-step problem

MOVIE RECORDING A film class is recording a DVD of student-made short films. Each student group is allotted up to 300 megabytes (MB) of video space. The films are encoded on the DVD at two different rates: a standard rate of 0.4 MB/sec for normal scenes and a high-quality rate of 1.2 MB/sec for complex scenes.

• Write an inequality describing the possible amounts of time available for standard and high-quality video.

• Graph the inequality.

• Identify three possible solutions of the inequality.

Solution

STEP 1 Write an inequality. First write a verbal model.

Standard rate

(MB/sec)

p

Standard time(sec)

1

High-quality rate

(MB/sec)

p

High-quality time (sec)

≤

Total space(MB)

0.4 p x 1 1.2 p y ≤ 300

An inequality is 0.4x 1 1.2y ≤ 300.

STEP 2 Graph the inequality. First graph the boundary line 0.4x 1 1.2y 5 300. Use a solid line because the inequality symbol is ≤.

Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the half-plane that contains (0, 0). Because x and y cannot be negative, shade only points in the first quadrant.

STEP 3 Identify solutions. Three solutions are given below and on the graph.

(150, 200) 150 seconds of standard and 200 seconds of high quality



For the first solution, 0.4(150) 1 1.2(200) 5 300, so all of the available space is used. For the other two solutions, not all of the space is used.

✓ GUIDED PRACTICE for Examples 2 and 3

Graph the inequality in a coordinate plane.

5. y > 21 6. x ≥ 24 7. y ≥ 23x

8. y < 2x 1 3 9. x 1 3y < 9 10. 2x 2 6y > 12

(300, 120)(600, 25)

(150, 200)

Standard (sec)

High

qua

lity

(sec

)

200 400 600 800

300

200

100

00 x

y



5. y > 21 6. x ≥ 24 7. y ≥ 23x

8. y < 2x 1 3 9. x 1 3y < 9 10. 2x 2 6y > 12


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ABSOLUTE VALUE INEQUALITIES Graphing an absolute value inequality is similar to graphing a linear inequality, but the boundary is an absolute value graph.


11. WHAT IF? Repeat the steps of Example 4 if each student group is allotted up to 420 MB of video space.


12. y ≤ x 2 2 1 1 13. y ≥ 2x 1 3 2 2 14. y < 3x 2 1 2 3

1. VOCABULARY Copy and complete: The graph of a linear inequality in two variables is a(n) ? .

2. ★ WRITING Compare the graph of a linear inequality in two variables with the graph of a linear equation in two variables.

CHECKING SOLUTIONS Tell whether the given ordered pairs are solutions of the inequality.

3. x > 27; (0, 10), (28, 25) 4. y ≤ 25x; (3, 2), (22, 1)

5. y ≥ 22x 1 4; (0, 4), (21, 8) 6. 2x 2 y < 3; (0, 0), (2, 22)

GRAPHING INEQUALITIES Graph the inequality in a coordinate plane.

7. x < 3 8. x ≥ 6 9. y > 22

10. 22y ≤ 8 11. y ≤ 22x 2 1 12. y < 3x 1 3

13. y > 3 } 4

x 1 1 14. y ≥ 2 2 } 3

x 2 2 15. 2x 1 y < 6

16. x 1 4y > 212 17. 3x 2 y ≥ 1 18. 2x 1 5y ≤ 210

2.8 EXERCISES

EXAMPLE 1

on p. 132for Exs. 3–6

EXAMPLES2 and 3


2.8 EXERCISES

EXAMPLE 1


E X A M P L E 5 Graph an absolute value inequality

Graph y > 22x 2 3 1 4 in a coordinate plane.

Solution

STEP 1 Graph the equation of the boundary, y 5 22x 2 3 1 4. Use a dashed line because the inequality symbol is >.

STEP 2 Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the portion of the coordinate plane outside the absolute value graph.

(0, 0)x

y

2

2

y > 22 z x 2 3 z 1 4

SKILL PRACTICE

HOMEWORKKEY

5 WORKED-OUT SOLUTIONSon p. WS1 for Exs. 15, 25, and 45

★ 5 STANDARDIZED TEST PRACTICEExs. 2, 21, 28, 39, 40, 41, 46, and 48


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ERROR ANALYSIS Describe and correct the error in graphing the inequality.

19. y < 2x 1 3 20. y ≥ 23x 2 2

21. ★ MULTIPLE CHOICE Which ordered pair is not a solution of 3x 2 5y < 30?

A (0, 0) B (21, 7) C (1, 27) D (25, 25)

ABSOLUTE VALUE INEQUALITIES Graph the inequality in a coordinate plane.

22. y > x 2 1 23. y < x 1 5 24. y > x 1 4 2 3

25. y ≤ 2 1 } 2

x 2 2 1 1 26. y < 3x 1 2 27. y ≥ 2x 2 1 2 4

28. ★ MULTIPLE CHOICE The graph of which inequality is shown?

A y ≤ 22x 1 1 1 3 B y ≥ 22x 2 1 1 3

C y > 22x 1 1 1 3 D y ≥ 22x 1 1 1 3

CHECKING SOLUTIONS Tell whether the given ordered pairs are solutions of the inequality.

29. y ≥ 2 2 } 3

x 1 1 } 2

; (26, 8), (23, 23) 30. 4.5 1 y < 1.6x; (0.5, 1), (3.8, 0)

31. 0.2x 1 0.7y > 21; (0.5, 21), (23, 21.5) 32. 1 } 4

x 2 y > 1; 1 4 } 3

, 0 2 , 1 2 } 3

, 24 2

GRAPHING INEQUALITIES Graph the inequality in a coordinate plane.

33. 3y < 4.5x 1 15 34. 21.5y 2 2x > 3 35. 2y 2 0.2 > 20.6x

36. 2 } 3

x 1 1 } 2

y > 2 37. y ≥ 2 5 } 2

x 2 3 2 3 } 2

38. 2y 2 4 ≤ 23x 1 2

39. ★ OPEN-ENDED MATH Write a linear inequality in two variables that has (21, 3) and (1, 6) as solutions, but does not have (4, 0) as a solution.

40. ★ WRITING Explain why it is not helpful when graphing a linear inequality in two variables to choose a test point that lies on the boundary line.

41. ★ SHORT RESPONSE Write an inequality for the graph shown. Explain how you came up with the inequality. Then describe a real-life situation that the first-quadrant portion of the graph could represent.

42. CHALLENGE Write an absolute value inequality that has exactly one solution in common with y ≥ 2x 2 3 1 5. The common solution should not be the vertex (3, 5) of the boundary. Explain how you found your inequality.

EXAMPLE 5

on p. 135for Exs. 22–28

5 WORKED-OUT SOLUTIONSon p. WS1

x

y

2

1

x

y

1

1

x

y

11 x

y

1

2

★ 5 STANDARDIZED TEST PRACTICE


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EXAMPLE 4

on p. 134for Exs. 43–48

43. CALLING CARDS You have a $20 phone card. Calls made using the card cost $.03 per minute to destinations within the United States and $.06 per minute to destinations in Brazil. Write an inequality describing the numbers of minutes you can use for calls to U.S. destinations and to Brazil.

44. RESTAURANT MANAGEMENT A pizza shop has 300 pounds (4800 ounces) of dough. A small pizza uses 12 ounces of dough and a large pizza uses 18 ounces of dough. Write and graph an inequality describing the possible numbers of small and large pizzas that can be made. Then give three possible solutions.

45. CRAFTS Cotton lace costs $1.50 per yard and linen lace costs $2.50 per yard. You plan to order at most $75 of lace for crafts. Write and graph an inequality describing how much of each type of lace you can order. If you buy 24 yards of cotton lace, what are the amounts of linen lace you can buy?

46. ★ SHORT RESPONSE You sell T-shirts for $15 each and caps for $10 each. Write and graph an inequality describing how many shirts and caps you must sell to exceed $1800 in sales. Explain how you can modify this inequality to describe how many shirts and caps you must sell to exceed $600 in profit if you make a 40% profit on shirts and a 30% profit on caps.

47. MULTI-STEP PROBLEM On a two week vacation, you and your brother can rent one canoe for $11 per day or rent two mountain bikes for $13 each per day. Together, you have $120 to spend.

a. Write and graph an inequality describing the possible numbers of days you and your brother can canoe or bicycle together.

b. Give three possible solutions of the inequality from part (a).

c. You decide that on one day you will canoe alone and your brother will bicycle alone. Repeat parts (a) and (b) using this new condition.

48. ★ EXTENDED RESPONSE While camping, you and a friend filter river water into two cylindrical containers with the radii and heights shown. You then use these containers to fill the water cooler shown.

a. Find the volumes of the containers and the cooler in cubic inches.

b. Using your results from part (a), write and graph an inequality describing how many times the containers can be filled and emptied into the water cooler without the cooler overflowing.

c. Convert the volumes from part (a) to gallons (1 in.3 ø 0.00433 gal). Then rewrite the inequality from part (b) in terms of these converted volumes.

d. Graph the inequality from part (c). Compare the graph with your graph from part (b), and explain why the results make sense.

PROBLEM SOLVING


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138

49. CHALLENGE A widescreen television image has a width w and a height h

that satisfy the inequality w } h

> 4 } 3

.

a. Does the television screen shown at the right meet the requirements of a widescreen image?

b. Let d be the length of a diagonal of a television image. Write an inequality describing the possible values of d and h for a widescreen image.

Graph the function. Compare the graph with the graph of y 5 x. (p. 123)

1. y 5 x 1 7 1 4 2. y 5 22x 1 10 2 1 3. f(x) 5 1 } 2

x 2 1 2 5

Write an equation of the graph. (p. 123)

4. x

y

121

5.

x

y

1

1

6.

x

y

2

28

Graph the inequality in a coordinate plane. (p. 132)

7. y > 22 8. y ≤ 3x 1 1 9. 2x 2 5y ≥ 10

10. MINI-CARS You have a 20 credit gift pass to a mini-car raceway. It takes 2 credits to drive the cars on the Rally track and 3 credits to drive the cars on the Grand Prix track. Write and graph an inequality describing how many times you can race on the two tracks using your gift pass. Then give three possible solutions. (p. 132)

QUIZ for Lessons 2.7–2.8

EXTRA PRACTICE for Lesson 2.8, p. 1011 ONLINE QUIZ at classzone.com

PREVIEW

Prepare for Lesson 3.1in Exs. 52–57.

MIXED REVIEW

Look for a pattern in the table. Then write an equation that represents the table. (p. 34)

50. x 0 1 2 3

y 11 15 19 23

51. x 0 1 2 3

y 60 45 30 15

Graph the equation. (p. 89)

52. x 1 3y 5 26 53. 4x 2 3y 5 15 54. 8x 2 6y 5 18

55. 6x 1 9y 5 18 56. 22x 2 5y 5 20 57. 210x 1 4y 5 20

Write an equation of the line that satisfies the given conditions. (p. 98)

58. m 5 4 } 5

, passes through (10, 22) 59. m 5 23, passes through (3, 7)

60. passes through (0, 2) and (5, 8) 61. passes through (4, 21) and (7, 26)


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2.8 Graph Linear Inequalities in Two Variablesboisemath.weebly.com/uploads/2/6/4/3/26433359/chap._2.8_notes.pdf · 2.8 Graph Linear Inequalities in Two Variables 133 KEY CONCEPT For