Lesson 2 11 Solving Systems of Linear Inequalities

Lesson 2. 11 Solving Systems of Linear Inequalities Concept: Represent and Solve Systems of Inequalities Graphically EQ: How do I represent the solutions of a system of inequalities? (Standard REI. 12) Vocabulary: Solutions region, Boundary lines (dashed or solid), Inclusive, Non-inclusive, Half plane, Test Point 1 2. 3. 2: Solving Systems of Linear Inequalities

Key Concepts • A system of inequalities is two or more inequalities in the same variables that work together. • The solution to a system of linear inequalities is the set of all points that make all the inequalities in the system true. • The solution region is the intersection of the half planes of the inequalities where they overlap (the darker shaded region). 2 2. 3. 2: Solving Systems of Linear Inequalities

Steps to Graphing a System of Linear Inequalities 1. Graph the first inequality as a linear equation. - Use a solid line for inclusive (≤ or ≥) - Use a dashed line for non-inclusive (< or >) 2. Shade the half plane above the y-intercept for (> and ≥). Shade the half plane below the y-intercept for (< and ≤). 3. Follow steps 1 and 2 for the second inequality. 4. The overlap of the two shaded regions represents the solutions to the system of inequalities. 5. Check your answer by picking a test point from the solutions region. If you get a true statement for both inequalities then your answer is correct. 2. 3. 2: Solving Systems of Linear Inequalities 3

Guided Practice - Example 1 Solve the following system of inequalities graphically: 4 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 1, continued 1. Graph the line y = -x + 10. Use a dashed line because the inequality is non-inclusive (greater than). 2. Shade the solution set. Since the symbol > was used we will shade above the yintercept. 5 2. 3. 2: Solving Systems of Linear Inequalities

6 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 1, continued 4. Find the solutions to the system. The overlap of the two shaded regions, which is darker, represents the solutions to the system: 5. Check your answer. Verify that (14, 2) is a solution to the system. Substitute it into both inequalities to see if you get a true statement for both. 7 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 1, continued ✔ 2. 3. 2: Solving Systems of Linear Inequalities 8

Guided Practice - Example 2 Solve the following system of inequalities graphically: 9 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 2, continued 1. Graph the line y = x – 10. Use a dashed line because the inequality is non-inclusive (greater than). 2. Shade the solution set. Since the symbol > was used we will shade above the yintercept. 10 2. 3. 2: Solving Systems of Linear Inequalities

11 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 2, continued 4. Find the solutions to the system. The overlap of the two shaded regions, which is darker, represents the solutions to the system: 5. Check your answer. Verify that (3, 3) is a solution to the system. Substitute it into both inequalities to see if you get a true statement for both. 12 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 2, continued ✔ 2. 3. 2: Solving Systems of Linear Inequalities 13

Guided Practice - Example 3 Solve the following system of inequalities graphically: 4 x + y ≤ 2 y ≥ -2 14 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 3, continued 1. Graph the line 4 x + y = 2. Use a solid line because the inequality is inclusive (less than or equal to). Change to slopeintercept form: y = -4 x + 2 2. Shade the solution set. Since the symbol ≤ was used we will shade below the yintercept. 15 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 3, continued 3. Graph the line y = -2 on the same coordinate plane. Use a solid line because the inequality is inclusive (greater than or equal to). Shade the solution set. Since the symbol ≥ was used we will shade above the yintercept. 16 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 3, continued 4. Find the solutions to the system. The overlap of the two shaded regions, which is darker, represents the solutions to the system: 4 x + y ≤ 2 y ≥ -2 5. Check your answer. Verify that (0, -1) is a solution to the system. Substitute it into both inequalities to see if you get a true statement for both. 17 2. 3. 2: Solving Systems of Linear Inequalities

Guided Practice: Example 3, continued ✔ 2. 3. 2: Solving Systems of Linear Inequalities 18

You Try! Graph the following system of inequalities 1. y ˃ -x – 2 y + 5 x ˂ 2 2. 19 2. 3. 2: Solving Systems of Linear Inequalities