12 6 Graphing Inequalities in Two Variables Warm

12 -6 Graphing Inequalities in Two Variables Warm Up Problem of the Day Lesson Presentation Course 3

Graphing Inequalities in 12 -6 Two Variables Warm Up Find each equation of direct variation, given that y varies directly with x. y = 6 x 1 x y = 2. x is 60 when y is 12. 5 3. y is 126 when x is 18. y = 7 x 1. y is 18 when x is 3. 4. x is 4 when y is 20. Course 3 y = 5 x

Graphing Inequalities in 12 -6 Two Variables 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 3

Graphing Inequalities in 12 -6 Two Variables Learn to graph inequalities on the coordinate plane. Course 3

Graphing Inequalities in 12 -6 Two Insert Lesson Title Here Variables Vocabulary boundary linear inequality Course 3

Graphing Inequalities in 12 -6 Two Variables 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 3

Graphing Inequalities in 12 -6 Two Variables Course 3

Graphing Inequalities in 12 -6 Two Variables 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 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 1 A: Graphing Inequalities Graph each inequality. y<x– 1 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. Then determine on which side of the line the solutions lie. (0, 0) Test a point not on the line. y<x– 1 ? 0<0– 1 ? 0 < – 1 Course 3 Substitute 0 for x and 0 for y.

Graphing Inequalities in 12 -6 Two Variables Helpful Hint 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. ” Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 1 A Continued 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) Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 1 B: Graphing Inequalities y 2 x + 1 First graph the boundary line y = 2 x + 1. Since points that are on the line are solutions of y 2 x + 1, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 2 x + 1 lie. (0, 4) Choose any point not on the line. y ≥ 2 x + 1 ? 4≥ 0+1 Course 3 Substitute 0 for x and 4 for y.

Graphing Inequalities in 12 -6 Two Variables Helpful Hint Any point on the line y = 2 x + 1 is a solution of y ≥ 2 x + 1 because the inequality symbol ≥ means “greater than or equal to. ” Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 1 B Continued Since 4 1 is true, (0, 4) is a solution of y 2 x + 1. Shade the side of the line that includes (0, 4) Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 1 C: Graphing Inequalities 2 y + 5 x < 6 First write the equation in slope-intercept form. 2 y + 5 x < 6 2 y < – 5 x + 6 y < – 5 x + 3 2 Subtract 5 x from both sides. Divide both sides by 2. Then graph the line y = – 5 x + 3. Since points that 2 are on the line are not solutions of y < – 5 x + 3, 2 make the line dashed. Then determine on which side of the line the solutions lie. Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 1 C Continued (0, 0) Choose any point not on the line. 5 y < – 2 x + 3 ? 0 < 0 + 3 Substitute 0 for x ? and 0 for y. 0<3 Since 0 < 3 is true, (0, 0) is a 5 solution of y < – 2 x + 3. Shade the side of the line that includes (0, 0). Course 3 (0, 0)

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 1 A Graph each inequality. y<x– 4 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) Test a point not on the line. y<x– 4 ? 0<0– 4 ? 0 < – 4 Course 3 Substitute 0 for x and 0 for y.

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 1 A Continued 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) Course 3

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 1 B y > 4 x + 4 First graph the boundary line y = 4 x + 4. Since points that are on the line are solutions of y 4 x + 4, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 4 x + 4 lie. (2, 3) Choose any point not on the line. y ≥ 4 x + 4 ? 3≥ 8+4 Course 3 Substitute 2 for x and 3 for y.

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 1 B Continued Since 3 12 is not true, (2, 3) is not a solution of y 4 x + 4. Shade the side of the line that does not include (2, 3) Course 3

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 1 C 3 y + 4 x 9 First write the equation in slope-intercept form. 3 y + 4 x 9 3 y – 4 x + 9 y – 4 x + 3 3 Subtract 4 x from both sides. Divide both sides by 3. Then graph the line y = – 4 x + 3. Since points that 3 are on the line are solutions of y – 4 x + 3, make 3 the line solid. Then determine on which side of the line the solutions lie. Course 3

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 1 C Continued (0, 0) Choose any point not on the line. y – 4 x + 3 3 ? 0 0 + 3 Substitute 0 for x ? and 0 for y. 0 3 Since 0 3 is not true, (0, 0) is not a solution of y – 4 x + 3. 3 Shade the side of the line that does not include (0, 0). Course 3 (0, 0)

Graphing Inequalities in 12 -6 Two Variables Additional Example 2: Career Application 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? First find the equation of the line that corresponds to the inequality. In 0 days the writer writes 0 pages. In 1 day the writer writes no 1 more than 7 pages. 2 Course 3 point (0, 0) point (1, 7. 5)

Graphing Inequalities in 12 -6 Two Variables Helpful Hint The phrase “no more” can be translated as less than or equal to. Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 2 Continued m = 7. 5 – 0 = 7. 5 1– 0 1 y 7. 5 x + 0 With two known points, find the slope. The y-intercept is 0. Graph the boundary line y = 7. 5 x. Since points on the line are solutions of y 7. 5 x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y 7. 5 x lie. Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 2 Continued (2, 2) Choose any point not on the line. y 7. 5 x ? 2 7. 5 2 Substitute 2 for x and 2 for y. ? 2 15 Since 2 15 is true, (2, 2) is a solution of y 7. 5 x. Shade the side of the line that includes point (2, 2). Course 3

Graphing Inequalities in 12 -6 Two Variables Additional Example 2 Continued 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. Course 3

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 2 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? First find the equation of the line that corresponds to the inequality. In 0 days the writer writes 0 pages. In 5 days the writer writes no more than 20 pages. Course 3 point (0, 0) point (5, 20)

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 2 Continued - 0 =20 = 4 m = 20 5 -0 5 With two known points, find the slope. y 4 x + 0 The y-intercept is 0. Graph the boundary line y = 4 x. Since points on the line are solutions of y 4 x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y 4 x lie. Course 3

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 2 Continued (5, 60) Choose any point not on the line. y 4 x ? 60 4 5 Substitute 5 for x and 60 for y. ? 60 20 Since 60 20 is not true, (5, 60) is not a solution of y 4 x. Shade the side of the line that does not include (5, 60). Course 3

Graphing Inequalities in 12 -6 Two Variables Check It Out: Example 2 Continued y 200 180 Pages 160 140 120 100 80 60 40 20 x 5 10 15 20 25 30 35 40 45 50 Days 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. Course 3

Graphing Inequalities in 12 -6 Two Variables Lesson Quiz Part I Graph each inequality. 1. y < – 1 x + 4 3 Course 3

Graphing Inequalities in 12 -6 Two Variables Lesson Quiz Part II 2. 4 y + 2 x > 12 Course 3

Graphing Inequalities in 12 -6 Two Variables Lesson Quiz: Part III Tell whether the given ordered pair is a solution of each inequality. 3. y < x + 15 (– 2, 8) yes 4. y 3 x – 1 (7, – 1) Course 3 no