5.7 Graph Linear Inequalities in Two Variables You will graph linear inequalities in two variables. Essential Question: How do you graph a linear inequality

5.7 Graph Linear Inequalities in Two Variables

• You will graph linear inequalities in two variables.

• Essential Question: How do you graph a linear inequality in two variables?

You will learn how to answer thisquestion by graphing a boundaryline and shading one half-plane.

Warm-Up Exercises

SOLUTION

EXAMPLE 1 Standardized Test Practice

Which ordered pair is not a solution of x – 3y ≤ 6?

A (0, 0) B (6, –1) C (10, 3) D (–1, 2)

Check whether each ordered pair is a solution of the inequality.

Test (0, 0): x – 3y ≤ 6

0 – 3(0) ≤ 6

Write inequality.

Substitute 0 for x and 0 for y.

Simplify.0 ≤ 6

Warm-Up ExercisesEXAMPLE 1 Standardized Test Practice

Test (6, –1): x – 3y ≤ 6

6 – 3(–1) ≤ 6 Substitute 6 for x and –1 for y.

Write inequality.

Simplify.

So, (0, 0) is a solution of x – 3y ≤ 6 but (6, –1) is not a solution.

ANSWER

The correct answer is B. A B C D

9 ≤ 6

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Tell whether the ordered pair is a solution of –x + 2y < 8.

1. (0, 0)

ANSWER solution


2. (0, 4)

ANSWER not a solution



3. (3, 5)

ANSWER solution


Warm-Up ExercisesEXAMPLE 2 Graph a linear inequality in two variables

Graph the inequality y > 4x – 3.

STEP 2

0 > 4(0) – 3?

Test (0, 0) in y > 4x – 3.

SOLUTION

Graph the equation y = 4x – 3. The inequality is >, so use a dashed line.

STEP 1

0 >–3


Shade the half-plane that contains (0, 0), because (0, 0) is a solution of the inequality.

STEP 3

Why do you test a point that is noton the boundary line?

• What is a point on the graph thatis not a solution of the inequality?


Graph the inequality x + 2y ≤ 0.

SOLUTION

STEP 1Graph the equation x + 2y = 0. The inequality is , so use a solid line.

STEP 2

Test (1, 0) in x + 2y ≤ 0.

1 ≤ 0

1 + 2(0) ≤ 0?


Shade the half-plane that does not contain (1, 0), because (1, 0) is not a solution of the inequality.

STEP 3

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

4. Graph the inequality x + 3y ≥ –1.

ANSWER

Warm-Up ExercisesEXAMPLE 4 Graph a linear inequality in one variables

Graph the inequality y –3.

SOLUTION

Graph the equation y = –3. The inequality is , so use a solid line.

STEP 1

STEP 2

Test (2, 0) in y –3.

You substitute only the y-coordinate, because the inequality does not have the variable x.

0 –3


Shade the half-plane that contains (2, 0), because (2, 0) is a solution of the inequality.

STEP 3

How is graphing a linear inequalityin one variable different fromgraphing a linear inequality in twovariables?

Does it matter which point youtest?


Graph the inequality x < –1.

SOLUTION

Graph the equation x = –1. The inequality is <, so use a dashed line.

STEP 1

STEP 2

Test (3, 0) in x < –1.

You substitute only the x-coordinate, because the inequality does not have the variable y.

3 < –1


Shade the half-plane that does not contain (3, 0), because (3, 0) is not a solution of the inequality.

STEP 3

Is the point (21, y ) a solution ofthe inequality? Explain.


5. Graph the inequality y > 1.

ANSWER


6. Graph the inequality y 3.

ANSWER


7. Graph the inequality x < –2.

ANSWER

Warm-Up ExercisesEXAMPLE 6Solve a multi-step problem

Job Earnings

You have two summer jobs at a youth center. You earn $8 per hour teaching basketball and $10 per hour teaching swimming. Let x represent the amount of time (in hours) you teach basketball each week, and let y represent the amount of time (in hours) you teach swimming each week. Your goal is to earn at least $200 per week.


• Write an inequality that describes your goal in terms of x and y.

• Graph the inequality.

• Give three possible combinations of hours that will allow you to meet your goal.

SOLUTION

Write a verbal model. Then write an inequality.

STEP 1


STEP 2

Graph the inequality 8x + 10y ≥ 200.

First, graph the equation 8x + 10y = 200 in Quadrant I. The inequality is ≥ , so use a solid line.


Next, test (5, 5) in 8x + 10y ≥ 200

8(5) + 10(5) ≥ 200

90 ≥ 200

STEP 3Choose three points on the graph, such as (13, 12), (14, 10), and (16, 9). The table shows the total earnings for each combination of hours.

Finally, shade the part of Quadrant I that does not contain (5, 5), because (5, 5) is not a solution of the inequality.



8. WHAT IF? In Example 6, suppose that next summer you earn $9 per hour teaching basketball and $12.50 per hour teaching swimming. Write and graph an inequality that describes your goal. Then give three possible combinations of hours that will help you meet your goal.

9x + 12.5y ≥ 200ANSWER

Sample answer: (8, 12), (12, 10), (16, 6)

Warm-Up Exercises5.7 Graph Linear Inequalities in Two Variables

• You will graph linear inequalities in two variables.

• Essential Question: How do you graph a linear inequality in two variables?• Graph the boundary line,

using a solid or a dashed line.• Use a test point and shade one of the two half-planes.

Graph the boundary line, using a dashed or solid line. Test a point not on the boundary line. If it is asolution, shade the half-plane that contains the test point. If it is not a solution, shade the half-plane that does not contain the test point.

Warm-Up ExercisesDaily Homework Quiz

Tell whether the ordered pair is a solution of the inequality.

1. x + 4y 16; (3, 3)

>–

ANSWER no

2. 3x – 2y < 20; (5, –2)

ANSWER yes

Warm-Up ExercisesDaily Homework Quiz

For a fundraiser, students offer a basic car wash for $2 and a deluxe for $5. They want to earn at least $100 per day. Write an inequality that describes the goal. Graph the inequality. Identify and interpret one combination that meets the goal.

3.

ANSWER

2x + 5y 100, where x represents basic washes and y represents deluxe washes; (40, 15); 40 basic washes and 15 deluxe washes.

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5.7 Graph Linear Inequalities in Two Variables You will graph linear inequalities in two variables. Essential Question: How do you graph a linear inequality