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
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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).
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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
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Guided Practice - Example 1
Solve the following system of inequalities graphically:
𝑦 > −𝑥 + 10
𝑦 <
1
5
𝑥 −
2
4
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 noninclusive (greater than).
2. Shade the solution set. Since the symbol >
was used we will shade above the yintercept.
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2.3.2: Solving Systems of Linear Inequalities
Guided Practice: Example 1, continued
3. Graph the line 𝒚 =
coordinate plane.
𝟏
𝒙
𝟐
−
𝟓
𝟒
on the same
Use a dashed line because the inequality is noninclusive (less than).
Shade the solution set. Since the symbol <
was used we will shade below the yintercept.
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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:
𝑦 > −𝑥 + 10
1
5
𝑦 < 𝑥 −
2
4
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.
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2.3.2: Solving Systems of Linear Inequalities
Guided Practice: Example 1, continued
20
18
16
14
12
10
8
6
4
2
-20 -18 -16 -14 -12 -10 -8
-6
-4
-2 0
-2
2
4
6
8
10
12
14
16
18
20
-4
-6
-8
-10
-12
2.3.2: Solving Systems of Linear Inequalities
✔
8
Guided Practice - Example 2
Solve the following system of inequalities graphically:
𝑦 > 𝑥 − 10
𝑦 > −3𝑥 +4
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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 noninclusive (greater than).
2. Shade the solution set. Since the symbol >
was used we will shade above the yintercept.
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2.3.2: Solving Systems of Linear Inequalities
Guided Practice: Example 2, continued
3. Graph the line 𝒚 = −𝟑𝒙 + 𝟒 on the same
coordinate plane.
Use a dashed line because the inequality is noninclusive (greater than).
Shade the solution set. Since the symbol >
was used we will shade above the yintercept.
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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:
𝑦 > 𝑥 − 10
𝑦 > −3𝑥 +4
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.
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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:
4x + y ≤ 2
y ≥ -2
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2.3.2: Solving Systems of Linear Inequalities
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 slopeintercept form: y = -4x + 2
2. Shade the solution set. Since the symbol ≤
was used we will shade below the yintercept.
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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.
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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:
4x + 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.
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2.3.2: Solving Systems of Linear Inequalities
Guided Practice: Example 3, continued
✔
2.3.2: Solving Systems of Linear Inequalities
18
1.
You Try!
Graph the following system of inequalities
x ≤ -3
2.
y ˃ -x – 2
5
y + 5x ˂ 2
𝑦 < 𝑥+2
3
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2.3.2: Solving Systems of Linear Inequalities