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

2. Graphing Inequalities in Two Variables 12-6 Graph each inequality. y < x – 1 Example 1: 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. 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 ?

3. Graphing Inequalities in Two Variables 12-6 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

4. Graphing Inequalities in Two Variables 12-6 Example 1 Continued (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).

5. Graphing Inequalities in Two Variables 12-6 y  2x + 1 Example 2: Graphing Inequalities 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. Substitute 0 for x and 4 for y. y ≥ 2x + 1 4 ≥ 0 + 1 ?

6. Graphing Inequalities in Two Variables 12-6 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

7. Graphing Inequalities in Two Variables 12-6 Example 2 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)

8. Graphing Inequalities in Two Variables 12-6 2y + 5x < 6 Example 3: Graphing Inequalities First write the equation in slope-intercept form. 2y < –5x + 6 2y + 5x < 6 y < – x + 3 5 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 5 2 Subtract 5x from both sides. Divide both sides by 2.

9. Graphing Inequalities in Two Variables 12-6 Example 3 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 + 3 5 2 0 < 0 + 3 ? 0 < 3 ? Substitute 0 for x and 0 for y. (0, 0)

10. Graphing Inequalities in Two Variables 12-6 Graph each inequality. y < x – 4 Example 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) y < x – 4 Test a point not on the line. Substitute 0 for x and 0 for y. 0 < 0 – 4 ? 0 < –4 ?

11. Graphing Inequalities in Two Variables 12-6 Example 4 Continued (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).

12. Graphing Inequalities in Two Variables 12-6 y > 4x + 4 Example 5 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 + 4 3 ≥ 8 + 4 ?

13. Graphing Inequalities in Two Variables 12-6 Example 5 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)

14. Graphing Inequalities in Two Variables 12-6 3y + 4x  9 Example 6 First write the equation in slope-intercept form. 3y  –4x + 9 3y + 4x  9 y  – x + 3 4 3 Subtract 4x from both sides. 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. 4 3

15. Graphing Inequalities in Two Variables 12-6 Example 6 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 + 3 4 3 0  0 + 3 ? 0  3 ? Substitute 0 for x and 0 for y. (0, 0)

16. Graphing Inequalities in Two Variables 12-6 Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 a week for spending money. How many weeks can Keith withdraw money in his account and still have at least $200 in at the end of summer? Example 7: Real World

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

18. Graphing Inequalities in Two Variables 12-6 Example 8: Real World A taxi charges a $1.75 flat rate fee in addition to $0.65 per mile. Katie has no more than $15 to spend. How many miles can Katie travel without going over what she has to spend?

19. Graphing Inequalities in Two Variables 12-6 Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

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

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

22. Graphing Inequalities in Two Variables 12-6 1. Identify the graph of the given inequality. 6y + 3x > 12 A. B. Lesson Quiz for Student Response Systems

23. Graphing Inequalities in Two Variables 12-6 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) Lesson Quiz for Student Response Systems