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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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Notes Over 2.6 Checking Solutions of Inequalities Check whether the given ordered pairs are solutions of the inequality.

Notes Over 2.6 Checking Solutions of Inequalities Check whether the given ordered pairs are solutions of the inequality.

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