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A4.e How Do I Graph The Solution Set of A Linear Inequality in Two Variables? Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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Warm Up Find each equation of direct variation, given that y varies directly with x. 1. y is 18 when x is x is 60 when y is y is 126 when x is x is 4 when y is 20. y = 6x y = 7x Course Graphing Inequalities in Two Variables y = 5x y = x 1 5

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Problem of the Day The circumference of a pizza varies directly with its diameter. If you graph that direct variation, what will the slope be? Course Graphing Inequalities in Two Variables

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Learn to graph inequalities on the coordinate plane. Course Graphing Inequalities in Two Variables

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Vocabulary boundary line linear inequality Insert Lesson Title Here Course Graphing Inequalities in Two Variables

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A graph of a linear equation separates the coordinate plane into three parts: the points on one side of the line, the points on the boundary line, and the points on the other side of the line. Course Graphing Inequalities in Two Variables

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Course Graphing Inequalities in Two Variables

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When the equality symbol is replaced in a linear equation by an inequality symbol, the statement is a linear inequality. Any ordered pair that makes the linear inequality true is a solution. Course Graphing Inequalities in Two Variables

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Graph each inequality. y < x – 1 Example 1A: Graphing Inequalities First graph the boundary line y = x – 1. Since no points that are on the line are solutions of y < x – 1, make the line dashed (creates 2 open half planes). Then determine on which side of the line the solutions lie. (0, 0) y < x – 1 Test a point not on the line. Substitute 0 for x and 0 for y. 0 < 0 – 1 ? 0 < –1 ? Course Graphing Inequalities in Two Variables

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Course Graphing Inequalities in Two Variables Any point on the line y = x 1 is not a solution of y < x 1 because the inequality symbol < means only less than and does not include equal to. Helpful Hint

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Example 1A Continued Course Graphing Inequalities in Two Variables (0, 0) Since 0 < –1 is not true, (0, 0) is not a solution of y < x – 1. Shade the side of the line that does not include (0, 0). When a < is used, shade below the boundary line.

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y 2x + 1 Additional Example 1B: Graphing Inequalities 1. Graph the boundary line: y = 2x Make the line solid (creates 2 closed half planes ) because the points that are on the line are solutions of y 2x Shade above the line (because the is used). This is where the rest of the solutions of y 2x + 1 lie. (0, 4) Choose any point not on the line. Substitute 0 for x and 4 for y. y 2x ? Course Graphing Inequalities in Two Variables

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Course Graphing Inequalities in Two Variables Any point on the line y = 2x 1 is a solution of y 2x 1 because the inequality symbol means greater than or equal to. Helpful Hint

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Additional Example 1B Continued Since 4 1 is true, (0, 4) is a solution of y 2x + 1. Shade the side of the line that includes (0, 4). When a > is used, shade above the boundary line. Course Graphing Inequalities in Two Variables (0, 4)

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2y + 5x < 6 Additional Example 1C: Graphing Inequalities First write the equation in slope-intercept form. 2y < –5x + 6 2y + 5x < 6 y < – x Then graph the line y = – x + 3. Since points that are on the line are not solutions of y < – x + 3, make the line dashed. Then determine on which side of the line the solutions lie Subtract 5x from both sides. Divide both sides by 2. Course Graphing Inequalities in Two Variables

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Additional Example 1C Continued Since 0 < 3 is true, (0, 0) is a solution of y < – x + 3. Shade the side of the line that includes (0, 0). 5 2 (0, 0)Choose any point not on the line. y < – x < ? 0 < 3 ? Course Graphing Inequalities in Two Variables Substitute 0 for x and 0 for y. (0, 0)

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Graph each inequality. y < x – 4 Check It Out: Example 1A First graph the boundary line y = x – 4. Since no points that are on the line are solutions of y < x – 4, make the line dashed. Then determine on which side of the line the solutions lie. (0, 0) y < x – 4 Test a point not on the line. Substitute 0 for x and 0 for y. 0 < 0 – 4 ? 0 < –4 ? Course Graphing Inequalities in Two Variables

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Check It Out: Example 1A Continued Course Graphing Inequalities in Two Variables (0, 0) Since 0 < –4 is not true, (0, 0) is not a solution of y < x – 4. Shade the side of the line that does not include (0, 0).

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y > 4x + 4 Check It Out: Example 1B First graph the boundary line y = 4x + 4. Since points that are on the line are solutions of y 4x + 4, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 4x + 4 lie. (2, 3) Choose any point not on the line. Substitute 2 for x and 3 for y. y 4x ? Course Graphing Inequalities in Two Variables

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Check It Out: Example 1B Continued Since 3 12 is not true, (2, 3) is not a solution of y 4x + 4. Shade the side of the line that does not include (2, 3). Course Graphing Inequalities in Two Variables (2, 3)

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3y + 4x 9 Check It Out: Example 1C First write the equation in slope-intercept form. 3y –4x + 9 3y + 4x 9 y – x Subtract 4x from both sides. Divide both sides by 3. Course Graphing Inequalities in Two Variables 4 3 Then graph the line y = – x + 3. Since points that are on the line are solutions of y – x + 3, make the line solid. Then determine on which side of the line the solutions lie. 4 3

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Check It Out: Example 1C Continued Since 0 3 is not true, (0, 0) is not a solution of y – x + 3. Shade the side of the line that does not include (0, 0). 4 3 (0, 0)Choose any point not on the line. y – x ? 0 3 ? Course Graphing Inequalities in Two Variables Substitute 0 for x and 0 for y. (0, 0)

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A successful screenwriter can write no more than seven and a half pages of dialogue each day. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write a 200-page screenplay in 30 days? Additional Example 2: Career Application First find the equation of the line that corresponds to the inequality. In 0 days the writer writes 0 pages. point (0, 0) point (1, 7.5) In 1 day the writer writes no more than 7 pages. 1 2 Course Graphing Inequalities in Two Variables

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Course Graphing Inequalities in Two Variables The phrase no more can be translated as less than or equal to. Helpful Hint

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Additional Example 2 Continued With two known points, find the slope. y 7.5 x + 0 The y-intercept is 0. Graph the boundary line y = 7.5x. Since points on the line are solutions of y 7.5x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y 7.5x lie. Course Graphing Inequalities in Two Variables m = 7.5 – 0 1 – == 7.5

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(2, 2) Choose any point not on the line. y 7.5x Substitute 2 for x and 2 for y. Since 2 15 is true, (2, 2) is a solution of y 7.5x. Shade the side of the line that includes point (2, 2). Additional Example 2 Continued ? 2 15 ? Course Graphing Inequalities in Two Variables

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The point (30, 200) is included in the shaded area, so the writer should be able to complete the 200 page screenplay in 30 days. Additional Example 2 Continued Course Graphing Inequalities in Two Variables

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A certain author can write no more than 20 pages every 5 days. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write 140 pages in 20 days? Check It Out: Example 2 First find the equation of the line that corresponds to the inequality. In 0 days the writer writes 0 pages. point (0, 0) point (5, 20) In 5 days the writer writes no more than 20 pages. Course Graphing Inequalities in Two Variables

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Check It Out: Example 2 Continued m = = 20 5 = 4 With two known points, find the slope. y 4x + 0 The y-intercept is 0. Graph the boundary line y = 4x. Since points on the line are solutions of y 4x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y 4x lie. Course Graphing Inequalities in Two Variables

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(5, 60) Choose any point not on the line. y 4x Substitute 5 for x and 60 for y. Since is not true, (5, 60) is not a solution of y 4x. Shade the side of the line that does not include (5, 60). Check It Out: Example 2 Continued ? ? Course Graphing Inequalities in Two Variables

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The point (20, 140) is not included in the shaded area, so the writer will not be able to write 140 pages in 20 days. Check It Out: Example 2 Continued x y \ 20 Pages Days Course Graphing Inequalities in Two Variables

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Graph each inequality. 1. y < – x Course Graphing Inequalities in Two Variables Lesson Quiz Part I

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2. 4y + 2x > 12 Course Graphing Inequalities in Two Variables Lesson Quiz Part II

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Tell whether the given ordered pair is a solution of each inequality. 3. y < x + 15 (–2, 8) 4. y 3x – 1 (7, –1) yes no Course Graphing Inequalities in Two Variables Lesson Quiz: Part III

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