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

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 ?

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

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).

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 ≥ ?

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

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)

Graphing Inequalities in Two Variables y + 5x < 6 Example 3: 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.

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 < ? 0 < 3 ? Substitute 0 for x and 0 for y. (0, 0)

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 ?

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).

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 ≥ ?

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)

Graphing Inequalities in Two Variables y + 4x  9 Example 6 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 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

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  ? 0  3 ? Substitute 0 for x and 0 for y. (0, 0)

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

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

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?

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

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

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

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

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