Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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

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? 

Learn to graph inequalities on the coordinate plane.

Vocabulary boundary line linear inequality

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.

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.

Additional Example 1A: 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 ? Substitute 0 for x and 0 for y. 0 < –1 ?

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

Additional Example 1A 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). (0, 0)

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

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

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). (0, 4)

Additional Example 1C: Graphing Inequalities 2y + 5x < 6 First write the equation in slope-intercept form. 2y + 5x < 6 2y < –5x + 6 Subtract 5x from both sides. y < – x + 3 5 2 Divide both sides by 2. 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. 5 2

Additional Example 1C Continued (0, 0) Choose any point not on the line. y < – x + 3 5 2 0 < 0 + 3 ? Substitute 0 for x and 0 for y. 0 < 3 ? 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)

Test a point not on the line. Check It Out: Example 1A 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 ? Substitute 0 for x and 0 for y. 0 < –4 ?

Check It Out: Example 1A 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). (0, 0)

Choose any point not on the line. Check It Out: Example 1B y > 4x + 4 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. y ≥ 4x + 4 3 ≥ 8 + 4 ? Substitute 2 for x and 3 for y.

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). (2, 3)

Check It Out: Example 1C 3y + 4x  9 First write the equation in slope-intercept form. 3y + 4x  9 3y  –4x + 9 Subtract 4x from both sides. y  – x + 3 4 3 Divide both sides by 3. 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.

Check It Out: Example 1C Continued (0, 0) Choose any point not on the line. y  – x + 3 4 3 (0, 0) 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  – x + 3. Shade the side of the line that does not include (0, 0). 4 3

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. point (0, 0) In 1 day the writer writes no more than 7 pages. 1 2 point (1, 7.5)

The phrase “no more” can be translated as less than or equal to. Helpful Hint

Additional Example 2 Continued With two known points, find the slope. m = 7.5 – 0 1 – 0 7.5 1 = = 7.5 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.

Additional Example 2 Continued (2, 2) Choose any point not on the line. y  7.5x 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  7.5x. Shade the side of the line that includes point (2, 2).

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.

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. point (0, 0) In 5 days the writer writes no more than 20 pages. point (5, 20)

Check It Out: Example 2 Continued 20 - 0 5 - 0 m = = 20 5 = 4 With two known points, find the slope. The y-intercept is 0. y  4x + 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.

Check It Out: Example 2 Continued (5, 60) Choose any point not on the line. y  4x 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  4x. Shade the side of the line that does not include (5, 60).

Check It Out: Example 2 Continued y 200 180 160 140 120 100 80 60 40\ 20 Pages 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.

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 32 32

Lesson Quiz Part I Graph each inequality. 1. y < – x + 4 1 3

Lesson Quiz Part II 2. 4y + 2x > 12

Lesson Quiz: Part III 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

Lesson Quiz for Student Response Systems 1. Identify the graph of the given inequality. 6y + 3x > 12 A. B. 36 36

Lesson Quiz for Student Response Systems 2. Tell which ordered pair is a solution of the inequality y < x + 12. A. (–3, 5) B. (–4, 12) C. (–5, 8) D. (–7, 9) 37 37