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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
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Lesson 2.11Solving Systems ofLinear 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
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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).
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2.3.2: Solving Systems of Linear Inequalities
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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
Guided Practice - Example 1Solve the following system of inequalities graphically:
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2.3.2: Solving Systems of Linear Inequalities
𝑦>−𝑥+10
𝑦<12 𝑥−
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Guided Practice: Example 1, continued1. 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.
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2.3.2: Solving Systems of Linear Inequalities
Guided Practice: Example 1, continued3. Graph the line on the same coordinate
plane. Use a dashed line because the inequality is non-inclusive (less than).
Shade the solution set. Since the symbol < was used we will shade below the y-intercept.
2.3.2: Solving Systems of Linear Inequalities
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Guided Practice: Example 1, continued4. 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.
2.3.2: Solving Systems of Linear Inequalities
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𝑦>−𝑥+10𝑦<12 𝑥−
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Guided Practice: Example 1, continued
2.3.2: Solving Systems of Linear Inequalities
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Guided Practice - Example 2Solve the following system of inequalities graphically:
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2.3.2: Solving Systems of Linear Inequalities
𝑦>𝑥−104
Guided Practice: Example 2, continued1. 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.
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2.3.2: Solving Systems of Linear Inequalities
Guided Practice: Example 2, continued3. Graph the line on the same coordinate
plane. Use a dashed line because the inequality is non-inclusive (greater than).
Shade the solution set. Since the symbol > was used we will shade above the y-intercept.
2.3.2: Solving Systems of Linear Inequalities
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Guided Practice: Example 2, continued4. 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.
2.3.2: Solving Systems of Linear Inequalities
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𝑦>𝑥−104
Guided Practice: Example 2, continued
2.3.2: Solving Systems of Linear Inequalities
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Guided Practice - Example 3Solve the following system of inequalities graphically:
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2.3.2: Solving Systems of Linear Inequalities
4x + y ≤ 2y ≥ -2
Guided Practice: Example 3, continued1. 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 + 2
2. Shade the solution set. Since the symbol ≤ was used we will shade below the y-intercept.
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2.3.2: Solving Systems of Linear Inequalities
Guided Practice: Example 3, continued3. 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.
2.3.2: Solving Systems of Linear Inequalities
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Guided Practice: Example 3, continued4. 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.
2.3.2: Solving Systems of Linear Inequalities
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4x + y ≤ 2y ≥ -2
Guided Practice: Example 3, continued
2.3.2: Solving Systems of Linear Inequalities
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You Try!Graph the following system of inequalities
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2.3.2: Solving Systems of Linear Inequalities
y ˃ -x – 2 y + 5x ˂ 2
x ≤ -3
1. 2.
