Solving Linear Inequalities 5-5 Solving Linear Inequalities Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Objective Graph and solve linear inequalities in two variables.

Example 1A: Identifying Solutions of Inequalities Tell whether the ordered pair is a solution of the inequality. (–2, 4); y < 2x + 1 y < 2x + 1 4 2(–2) + 1 4 –4 + 1 4 –3 <  Substitute (–2, 4) for (x, y). (–2, 4) is not a solution.

Example 1B: Identifying Solutions of Inequalities Tell whether the ordered pair is a solution of the inequality. (3, 1); y > x – 4 y > x − 4 1 3 – 4 1 – 1 > Substitute (3, 1) for (x, y).  (3, 1) is a solution.

Graphing Linear Inequalities Step 1 Solve the inequality for y (slope-intercept form). Step 2 Graph the boundary line. Use a solid line for ≤ or ≥. Use a dashed line for < or >. Step 3 Shade the half-plane above the line for y > or ≥. Shade the half-plane below the line for y < or y ≤. Check your answer.

Example 2A: Graphing Linear Inequalities in Two Variables Graph the solutions of the linear inequality. y  2x – 3 Step 1 The inequality is already solved for y. Step 2 Graph the boundary line y = 2x – 3. Use a solid line for . Step 3 The inequality is , so shade below the line.

 Example 2A Continued Graph the solutions of the linear inequality. y  2x – 3 Substitute (0, 0) for (x, y) because it is not on the boundary line. Check y  2x – 3 0 2(0) – 3 0 –3   A false statement means that the half-plane containing (0, 0) should NOT be shaded. (0, 0) is not one of the solutions, so the graph is shaded correctly.

The point (0, 0) is a good test point to use if it does not lie on the boundary line. Helpful Hint

Example 2B: Graphing Linear Inequalities in Two Variables Graph the solutions of the linear inequality. 5x + 2y > –8 Step 1 Solve the inequality for y. 5x + 2y > –8 –5x –5x 2y > –5x – 8 y > x – 4 Step 2 Graph the boundary line Use a dashed line for >. y = x – 4.

Example 2B Continued Graph the solutions of the linear inequality. 5x + 2y > –8 Step 3 The inequality is >, so shade above the line.

 Example 2B Continued Graph the solutions of the linear inequality. 5x + 2y > –8 Substitute ( 0, 0) for (x, y) because it is not on the boundary line. Check y > x – 4 0 (0) – 4 0 –4 >  The point (0, 0) satisfies the inequality, so the graph is correctly shaded.

Example 2C: Graphing Linear Inequalities in two Variables Graph the solutions of the linear inequality. 4x – y + 2 ≤ 0 Step 1 Solve the inequality for y. 4x – y + 2 ≤ 0 –y ≤ –4x – 2 –1 –1 y ≥ 4x + 2 Step 2 Graph the boundary line y ≥= 4x + 2. Use a solid line for ≥.

Example 2C Continued Graph the solutions of the linear inequality. 4x – y + 2 ≤ 0 Step 3 The inequality is ≥, so shade above the line.

 Example 2C Continued Check y ≥ 4x + 2 3 4(–3)+ 2 3 –12 + 2 3 ≥ –10 3 4(–3)+ 2 3 –12 + 2 3 ≥ –10  y ≥ 4x + 2 Substitute ( –3, 3) for (x, y) because it is not on the boundary line. The point (–3, 3) satisfies the inequality, so the graph is correctly shaded.

Check It Out! Example 2a Graph the solutions of the linear inequality. 4x – 3y > 12 Step 1 Solve the inequality for y. 4x – 3y > 12 –4x –4x –3y > –4x + 12 y < – 4 Step 2 Graph the boundary line y = – 4. Use a dashed line for <.

Check It Out! Example 2a Continued Graph the solutions of the linear inequality. 4x – 3y > 12 Step 3 The inequality is <, so shade below the line.

Check It Out! Example 2a Continued Graph the solutions of the linear inequality. 4x – 3y > 12 Check y < – 4 –6 (1) – 4 –6 – 4 –6 <  Substitute ( 1, –6) for (x, y) because it is not on the boundary line. The point (1, –6) satisfies the inequality, so the graph is correctly shaded.