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Warm-Up Exercises 1. Solve |x – 6| = 4. 2. Solve |x + 5| – 8 = 2. ANSWER 2, 10 ANSWER –15, 5 3. A frame will hold photographs that are 5 inches by 8 inches with an absolute derivation of 0.25 inch for length and width. What are the minimum and maximum dimensions for photos? ANSWER min: 4.75 in. by 7.75 in. ; max: 5.25 in. by 8.25 in. Warm-Up 6.4 Students will be able to solve inequalities that have two variables.

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Warm-Up Exercises Homework Review

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Warm-Up Exercises Daily Homework Quiz For use after Lesson 6.5 Solve the equation. If possible. ANSWER – 9, 17 ANSWERno solutions 1. x– 4 = 132. x + 2 + 7 = 33. 2x– 6 + 4 = 20 ANSWER – 5, 11

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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

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Warm-Up Exercises EXAMPLE 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

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Warm-Up Exercises GUIDED PRACTICE for Example 1 SOLUTION Tell whether the ordered pair is a solution of – x + 2y < 8. Check whether each ordered pair is a solution of the inequality. Test (0, 0 ) – x + 2y < 8. 0 + 2(0) < 8 Write inequality. Substitute 0 for x and 0 for y. Simplify. 1. (0, 0) 0 < 8

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Warm-Up Exercises GUIDED PRACTICE for Example 1 ANSWER So, (0, 0) is a solution of – x + 2y < 8.

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Warm-Up Exercises SOLUTION GUIDED PRACTICE for Example 1 2. (0, 4) Check whether each ordered pair is a solution of the inequality. Test (0, 4 ) – x + 2y < 8. 0 + 2(4) < 8 Write inequality. Substitute 0 for x and 4 for y. Simplify. 0 + 8 < 8 ANSWER So, (0, 4) is not a solution of – x + 2y < 8.

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Warm-Up Exercises GUIDED PRACTICE for Example 1 SOLUTION 3. (3, 5) Check whether each ordered pair is a solution of the inequality. Test (3, 5 ) – x + 2y < 8. – 3 + 2(5) < 8 Write inequality. Substitute – 3 for x and 5 for y. Simplify. 7 < 8 ANSWER So, (0, 0) is a solution of – x + 2y < 8.

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Warm-Up Exercises 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 0 > 4(0) – 3 ? Test (0, 0) in y > 4x – 3. 0 >–3

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Warm-Up Exercises EXAMPLE 2 Graph a linear inequality in two variables Shade the half-plane that contains (0, 0), because (0, 0) is a solution of the inequality. STEP 3

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Warm-Up Exercises 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 ≤ 01 1 + 2(0) ≤ 0 ?

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

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Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 4. Graph the inequality x + 3y ≥ –1 SOLUTION STEP 1 Graph the equation x + 3y = –1. The inequality is <, so use a solid line. STEP 2 Test (1, 0) in x + 3y ≤ –1 1 + 2(0) ≤ –1 ? 1

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Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 STEP 3 Shade the half-plane that contain (1, 0), because (1, 0) is a solution of the inequality.

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Warm-Up Exercises EXAMPLE 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

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Warm-Up Exercises EXAMPLE 4 Graph a linear inequality in one variables Shade the half-plane that contains (2, 0), because (2, 0) is a solution of the inequality. STEP 3

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Warm-Up Exercises EXAMPLE 5 Graph a linear inequality in one variables 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

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Warm-Up Exercises EXAMPLE 5 Graph a linear inequality in one variables Shade the half-plane that does not contains 3, 0), because (3, 0) is not a solution of the inequality. STEP 3

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Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 5. Graph the inequality y > 1. SOLUTION Graph the equation y = 1. The inequality is <, so use a dashed line. STEP 1 STEP 2 You substitute only the y -coordinate, because the inequality does not have the variable x. Test (1, 0) in y < 1. 1> 1

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Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 STEP 3 Shade the half-plane that contains (1, 0), because (1, 0) is a solution of the inequality.

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Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 SOLUTION Graph the equation y = 3. The inequality is <, so use a dashed line. STEP 1 STEP 2 You substitute only the y -coordinate, because the inequality does not have the variable x. Test (3, 0) in y < 3. 3> 3 6. Graph the inequality y < 3.

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Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 STEP 3 Shade the half-plane that contains (3, 0), because (3, 0) is a solution of the inequality.

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Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 SOLUTION Graph the equation y = –2. The inequality is <, so use a dashed line. STEP 1 STEP 2 You substitute only the y -coordinate, because the inequality does not have the variable x. Test (2, 0) in y < – 2. 7. Graph the inequality x < – 2. 2 <–2

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Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 STEP 3 Shade the half-plane that does not contains ( 2, 0), because (2, 0) is not a solution of the inequality.

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Warm-Up Exercises 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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Warm-Up Exercises 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 Write a verbal model. Then write an inequality. STEP 1

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Warm-Up Exercises EXAMPLE 6 Solve a multi-step problem 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.

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

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Warm-Up Exercises 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. SOLUTION Write a verbal model. Then write an inequality. STEP 1

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Warm-Up Exercises GUIDED PRACTICE for Example 6 STEP 2 Graph the inequality 9x + 12.5y ≥ 200 First, graph the equation 9x + 12.5y = 200 in Quadrant I. The inequality is ≥, so use a solid line.

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Warm-Up Exercises GUIDED PRACTICE for Example 6 Next, test (5, 5) in 9x + 12.5y ≥ 200 9(5) + 12.5(5) ≥ 200 107.5 ≥ 200 Finally, shade the part of Quadrant I that does not contain (5, 5), because (5, 5) is not a solution of the inequality.

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Warm-Up Exercises GUIDED PRACTICE for Example 6 STEP 3 Choose three points on the graph, such as (8, 12), (12, 10), and (16, 9). The table shows the total earnings for each combination of hours.

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