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.

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.

Solving Systems of 5-6 Linear Inequalities Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Objective Graph and solve systems of linear inequalities in two variables.

Example 1A: Identifying Solutions of Systems of Linear Inequalities Tell whether the ordered pair is a solution of the given system. y ≤ –3x + 1 (–1, –3); y < 2x + 2 (–1, –3) (–1, –3) y ≤ –3x + 1 y < 2x + 2 –3 –3(–1) + 1 –3 3 + 1 –3 4 ≤  –3 –2 + 2 –3 0 <  –3 2(–1) + 2 (–1, –3) is a solution to the system because it satisfies both inequalities.

Example 1B: Identifying Solutions of Systems of Linear Inequalities Tell whether the ordered pair is a solution of the given system. y < –2x – 1 (–1, 5); y ≥ x + 3 (–1, 5) (–1, 5) y < –2x – 1 5 2 ≥  5 –1 + 3 y ≥ x + 3 5 –2(–1) – 1 5 2 – 1 5 1 <  (–1, 5) is not a solution to the system because it does not satisfy both inequalities.

An ordered pair must be a solution of all inequalities to be a solution of the system. Remember!

Example 2B: Solving a System of Linear Inequalities by Graphing Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. –3x + 2y ≥ 2 y < 4x + 3 –3x + 2y ≥ 2 Solve the first inequality for y. 2y ≥ 3x + 2

(0, 0) and (–4, 5) are not solutions. Example 2B Continued Graph the system.  (2, 6) (1, 3) y < 4x + 3  (0, 0) (–4, 5) (2, 6) and (1, 3) are solutions. (0, 0) and (–4, 5) are not solutions.

Check It Out! Example 2b Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. y > x – 7 3x + 6y ≤ 12 3x + 6y ≤ 12 Solve the second inequality for y. 6y ≤ –3x + 12 y ≤ x + 2

Check It Out! Example 2b Continued Graph the system. y > x − 7 y ≤ – x + 2  (4, 4) (1, –6)  (0, 0) (3, –2) (0, 0) and (3, –2) are solutions. (4, 4) and (1, –6) are not solutions.

Example 3A: Graphing Systems with Parallel Boundary Lines Graph the system of linear inequalities. Describe the solutions. y ≤ –2x – 4 y > –2x + 5 This system has no solutions.

Example 3B: Graphing Systems with Parallel Boundary Lines Graph the system of linear inequalities. Describe the solutions. y > 3x – 2 y < 3x + 6 The solutions are all points between the parallel lines but not on the dashed lines.

Example 3C: Graphing Systems with Parallel Boundary Lines Graph the system of linear inequalities. Describe the solutions. y ≥ 4x + 6 y ≥ 4x – 5 The solutions are the same as the solutions of y ≥ 4x + 6.

Check It Out! Example 3b Graph the system of linear inequalities. Describe the solutions. y ≥ 4x – 2 y ≤ 4x + 2 The solutions are all points between the parallel lines including the solid lines.

Check It Out! Example 3c Graph the system of linear inequalities. Describe the solutions. y > –2x + 3 y > –2x The solutions are the same as the solutions of y > –2x + 3.