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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 in two variables? You will learn how to answer this question by graphing a boundary line and shading one half-plane.

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**Standardized Test Practice**

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) SOLUTION Check whether each ordered pair is a solution of the inequality. Test (0, 0): x – 3y ≤ 6 Write inequality. 0 – 3(0) ≤ 6 Substitute 0 for x and 0 for y. 0 ≤ 6 Simplify.

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**Standardized Test Practice**

EXAMPLE 1 Standardized Test Practice Test (6, –1): x – 3y ≤ 6 Write inequality. 6 – 3(–1) ≤ 6 Substitute 6 for x and –1 for y. 9 ≤ 6 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

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GUIDED PRACTICE for Example 1 Tell whether the ordered pair is a solution of –x + 2y < 8. 1. (0, 0) ANSWER solution

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GUIDED PRACTICE for Example 1 Tell whether the ordered pair is a solution of –x + 2y < 8. 2. (0, 4) ANSWER not a solution

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GUIDED PRACTICE for Example 1 Tell whether the ordered pair is a solution of –x + 2y < 8. 3. (3, 5) ANSWER solution

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EXAMPLE 2 Graph a linear inequality in two variables Graph the inequality y > 4x – 3. SOLUTION Graph the equation y = 4x – 3. The inequality is >, so use a dashed line. STEP 1 STEP 2 Test (0, 0) in y > 4x – 3. 0 > 4(0) – 3 ? 0 >–3

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EXAMPLE 2 Graph a linear inequality in two variables STEP 3 Shade the half-plane that contains (0, 0), because (0, 0) is a solution of the inequality. Why do you test a point that is not on the boundary line? • What is a point on the graph that is not a solution of the inequality?

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EXAMPLE 3 Graph a linear inequality in two variables Graph the inequality x + 2y ≤ 0. SOLUTION STEP 1 Graph the equation x + 2y = 0. The inequality is , so use a solid line. STEP 2 Test (1, 0) in x + 2y ≤ 0. 1 + 2(0) ≤ 0 ? 1 ≤ 0

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EXAMPLE 3 Graph a linear inequality in two variables STEP 3 Shade the half-plane that does not contain (1, 0), because (1, 0) is not a solution of the inequality.

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GUIDED PRACTICE for Examples 2 and 3 4. Graph the inequality x + 3y ≥ –1. ANSWER

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EXAMPLE 4 Graph a linear inequality in one variables Graph the inequality y –3. SOLUTION STEP 1 Graph the equation y = –3. The inequality is , so use a solid line. 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

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EXAMPLE 4 Graph a linear inequality in one variables STEP 3 Shade the half-plane that contains (2, 0), because (2, 0) is a solution of the inequality. How is graphing a linear inequality in one variable different from graphing a linear inequality in two variables? Does it matter which point you test?

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EXAMPLE 5 Graph a linear inequality in one variables Graph the inequality x < –1. SOLUTION STEP 1 Graph the equation x = –1. The inequality is <, so use a dashed line. 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

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EXAMPLE 5 Graph a linear inequality in one variables STEP 3 Shade the half-plane that does not contain (3, 0), because (3, 0) is not a solution of the inequality. Is the point (21, y ) a solution of the inequality? Explain.

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GUIDED PRACTICE for Examples 4 and 5 5. Graph the inequality y > 1. ANSWER

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GUIDED PRACTICE for Examples 4 and 5 6. Graph the inequality y 3. ANSWER

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GUIDED PRACTICE for Examples 4 and 5 7. Graph the inequality x < –2. ANSWER

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EXAMPLE 6 Solve 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.

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EXAMPLE 6 Solve a multi-step problem • 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 STEP 1 Write a verbal model. Then write an inequality.

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EXAMPLE 6 Solve a multi-step problem STEP 2 First, graph the equation 8x + 10y = 200 in Quadrant I. The inequality is ≥ , so use a solid line. Graph the inequality 8x + 10y ≥ 200.

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EXAMPLE 6 Solve a multi-step problem Next, test (5, 5) in 8x + 10y ≥ 200 : Finally, shade the part of Quadrant I that does not contain (5, 5), because (5, 5) is not a solution of the inequality. 8(5) + 10(5) ≥ 200 90 ≥ 200 STEP 3 Choose 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.

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EXAMPLE 6 Solve a multi-step problem

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GUIDED PRACTICE for Example 6 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 y ≥ 200 ANSWER Sample answer: (8, 12), (12, 10), (16, 6)

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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 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 a solution, 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.

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Daily Homework Quiz Tell whether the ordered pair is a solution of the inequality. x + 4y ; (3, 3) > – ANSWER no x – 2y < 20; (5, –2) ANSWER yes

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Daily 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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Graph linear inequalities with one variable EXAMPLE 2 Graph ( a ) y < – 3 and ( b ) x < 2 in a coordinate plane. Test the point (0,0). Because (0,0) is.

Graph linear inequalities with one variable EXAMPLE 2 Graph ( a ) y < – 3 and ( b ) x < 2 in a coordinate plane. Test the point (0,0). Because (0,0) is.

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