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 2.3.2: Solving Systems of Linear Inequalities 1

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) : Solving Systems of Linear Inequalities

3 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 ( ) 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 : Solving Systems of Linear Inequalities

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

Guided Practice: Example 1, continued 1.Graph the line y = -x 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 y- intercept : Solving Systems of Linear Inequalities

6

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 : Solving Systems of Linear Inequalities 7

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: : 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 y- intercept : Solving Systems of Linear Inequalities

11

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 : Solving Systems of Linear Inequalities 12

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: : Solving Systems of Linear Inequalities 4x + y ≤ 2 y ≥ -2

Guided Practice: Example 3, continued 1.Graph the line 4x + y = 2. Use a solid line because the inequality is inclusive (less than or equal to). Change to slope- intercept form: y = -4x Shade the solution set. Since the symbol ≤ was used we will shade below the y- intercept : 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 y- intercept : Solving Systems of Linear Inequalities 16

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: 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 : Solving Systems of Linear Inequalities 17 4x + y ≤ 2 y ≥ -2

Guided Practice: Example 3, continued 2.3.2: Solving Systems of Linear Inequalities 18 ✔

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