6.7. Graph Linear Inequalities in Two Variables. Warm Up Lesson Presentation Lesson Quiz

Transcription

1 6.7 Graph Linear Inequalities in Two Variables Warm Up Lesson Presentation Lesson Quiz

2 6.7 Warm-Up Tell whether the ordered pair is a solution of the equation. 1. x + 2y = 4; (2, 1) no 2. 4x + 3y = 22; (7, 2) yes

3 6.7 Warm-Up 3. Graph the equation y 2x = 4.

4 6.7 Example 1 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 0 3(0) Write inequality. Substitute 0 for x and 0 for y. Simplify.

5 6.7 Example 1 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. The correct answer is B. A B C D

6 6.7 Guided Practice Tell whether the ordered pair is a solution of x + 2y < (0, 0) solution 2. (0, 4) not a solution 3. (3, 5) solution

7 6.7 Example 2 Graph the inequality y > 4x 3. SOLUTION STEP 1 Graph the equation y = 4x 3. The inequality is >, so use a dashed line. STEP 2 Test (0, 0) in y > 4x 3.? 0 > 4(0) 3 0 > 3 STEP 3 Shade the half-plane that contains (0, 0), because (0, 0) is a solution of the inequality.

8 6.7 Example 3 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) STEP 3 Shade the half-plane that does not contain (1, 0), because (1, 0) is not a solution of the inequality.

9 6.7 Guided Practice 4. Graph the inequality x + 3y 1.

10 6.7 Example 4 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 STEP 3 Shade the half-plane that does not contain (3, 0), because (3, 0) is not a solution of the inequality.

11 6.7 Guided Practice 5. Graph the inequalities (a) y > 1 and (b) x < 2. (a) (b)

12 6.7 Example 5 Write an inequality represented by the graph. SOLUTION STEP 1 Note from the graph that the slope of the boundary line is 2 and the y-intercept is 2. STEP 2 Write and equation in slope-intercept form, y = mx + b. y = 2x + ( 2) = 2x 2 STEP 3 Write an inequality. The boundary line is dashed, so replace = with either < or >. The graph is shaded below the boundary line, so choose <. The inequality represented by the graph is y < 2x 2.

13 6.7 Guided Practice 6. WHAT IF? In Example 5, suppose the boundary line were solid. What inequality would be represented by the graph? y 2x 2

14 6.7 Example 6 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.

15 6.7 Example 6 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.

16 6.7 Example 6 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.

17 6.7 Example 6 Next, test (5, 5) in 8x + 10y 200 : 8(5) + 10(5) Finally, shade the part of Quadrant I that does not contain (5, 5), because (5, 5) is not a solution of the inequality.

18 6.7 Example 6 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.

19 6.7 Guided Practice 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 Sample answer: (8, 12), (12, 10), (16, 6)

20 6.7 Lesson Quiz Tell whether the ordered pair is a solution of the inequality. 1. x + 4y > 16; (3, 3) no 2. 3x 2y < 20; (5, 2) yes

21 6.7 Lesson Quiz 3. 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. 2x + 5y 100, where x represents basic washes and y represents deluxe washes; (40, 15); 40 basic washes and 15 deluxe washes.