Name Class Date 12.2Graphing Systems of Linear
Inequalities
Essential Question: How do you solve a system of linear inequalities?
Resource
Locker
Explore Determining Solutions of Systems
of Linear Inequalities
A system of linear inequalities consists of two or more linear inequalities that have the same variables. The solutions
of a system of linear inequalities are all the ordered pairs that make all the inequalities in the system true.
Solve the system of equations by graphing.
⎧  x + 3y > 3
     
​
⎨ 
 ​ ​​  
⎩ -x + y ≤ 6
A
First look at x + 3y > 3. The equation of the boundary line is .
B
What are the x-and y-intercepts?
C
The inequality symbol is > so use a line.
DShade the boundary line for solutions that are greater than the inequality.
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EGraph x + 3y > 3.
8
y
4
x
-8
-4
0
-4
4
8
-8
F
Look at -x + y ≤ 6. The equation of the boundary line is .
G
What are the x-and y-intercepts?
H
The inequality symbol is ≤ so use a line.
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Lesson 2
IShade the boundary line for solutions that are less than the inequality.
JGraph -x + y ≤ 6 on the same graph as x + 3y > 3.
y
8
4
x
-8
-4
0
-4
4
8
-8
K
Identify the solutions. They are represented by the shaded regions.
L
Check your answer by using a point in each region. Complete the table.
Ordered Pair
Satisfies x + 3y > 3?
Satisfies -x + y ≤ 6?
In the overlapping
shaded regions?
​(0,  0)​
​(2,  3)​
​(-8, 2)​
​(-4, 6)​
Reflect
1.
Discussion Why is (​ 0, 0)​a good point to use for checking the answer to this system of linear inequalities?
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Lesson 2
Explain 1
Solving Systems of Linear Inequalities by Graphing
You can use a graph of a system of linear inequalities to determine and identify solutions to the system
of linear inequalities.
Graph the system of linear inequalities. Give two ordered pairs that are solutions
and two that are not solutions.
Example 1
⎧-6x + 3y ≤ 12
A ⎨ _1
y> x-3
8
⎩
2
Solve the first inequality for y.
Graph the system.
-6x + 3y ≤ 12
⎧y ≤ 2x + 4
⎨
3y ≤ 6x + 12
4
x
-8
-4
1
2
⎩ y > _x - 3
y ≤ 2x + 4
0
⎧3x + y ≤ 1
8
⎩
4
B ⎨ _2
y> x-2
3
Graph the system.
3x + y ≤ 1
⎧
y≤
-8
-4
2
_
⎩y > 3 x - 2
and
are solutions.
8
y
x
y≤
⎨
4
-4
-8
(0, 0) and (2, 8) are solutions. (-6, -4) and (-4, 4) are not solutions.
Solve the first inequality for y.
y
0
4
-4
8
-8
and
are not solutions.
Reflect
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2.
Is (–6, –6) a solution of the system?
Your Turn
Graph the system of linear inequalities. Give two ordered pairs that are solutions and
two that are not solutions.
3.
⎧y ≤ x + 3
⎨
⎩ y < -3
8
y
4.
4
⎧ y>x-8
⎨
⎩ 2x + 4y < 16
8
y
4
x
-8
-4
0
-4
4
x
-8
8
-8
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-4
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Lesson 2
Graphing Systems of Inequalities with Parallel
Boundary Lines
Explain 2
If the lines in a system of linear equations are parallel, there are no solutions. However, if the boundary lines in a
system of linear inequalities are parallel, the system may or may not have solutions.
Example 2
A
Graph each system of linear inequalities. Describe the solutions.
⎧y < 4x - 3
⎨
⎩ y > 4x + 2
B
⎧y > x - 2
⎨
⎩y ≤ x + 4
y
8
y
8
4
4
x
x
-8
-4
0
4
-4
-8
8
-4
0
4
-4
8
-8
-8
This system has no solution.
The solutions are all points
the parallel lines
and on the
line.
Your Turn
Graph each system of linear inequalities. Describe the solutions.
1x - 6
⎧y < _
⎧y ≤ -2x - 3
3
6. ⎨
5. ⎨
1
⎩ y ≤ -2x + 1
⎩ y ≥ _x + 5
3
8
y
8
4
x
-8
-4
0
-4
4
x
-8
8
-8
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4
y
-4
0
-4
4
8
-8
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Lesson 2
Elaborate
7.
Is it possible for a system of two linear inequalities to have every point in the plane as solutions? Why or
why not?
8.
Discussion How would you write a system of linear inequalities from a graph?
9.
Essential Question Check-In How does testing specific ordered pairs tell you that the solution you
graphed is correct?
Evaluate: Homework and Practice
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1.
Match the inequality with the correct boundary line. Answers may be used more than once.
a. y = 3x
-x + 3y ≤ 0
1x
b. y = _
3
1
d. y = -x + _
2
1
y > -x + _
2
1x
y≤_
3
2 + 1y ≥ x
_
3
3
e. y = 3x - 2
-y > x - 0.5
f. y = x
1y ≥ x
_
3
c. y = x - 0.5
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• Hints and Help
• Extra Practice
_
551
Lesson 2
Determine if the given point satisfies either equation and is a solution of the system of inequalities.
2.
 ⎧4y - 20x < 6
​ ​​      
  
   ​; ​(0, 0)​
 _
5
​ 
⎩ 2 ​  y ≥ 5x - 10
⎨ 
3.
 ⎧x + 5y > -10
     
 ​; ​(2.5,  -1.5)​
⎨ 
 ​ ​​    
⎩x - y ≤ 4
Determine if the given point is a solution of the system of inequalities. If not, find a point that is.
4.
(​-9, 4)​
5.
8
(​ 6, -2)​
y
y
8
4
4
x
-8
-4
0
-4
4
x
-8
8
-8
6.
-4
0
-4
4
8
-8
​(0, -4)​
8
y
4
x
-8
-4
0
4
-4
8
Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not
solutions.
7.
 ⎧x > 2
​
​ ​​      
  
 
 
1
⎩ y ≤ -​ _ ​ x - 2
2
⎨ 
8
y
8.
4
 ⎧y > -x
   
   
​
⎨ 
 ​ ​​ 
⎩y ≥ x
8
y
4
x
x
-8
-4
0
-4
4
-8
8
0
-4
4
8
-8
-8
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Lesson 2
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-8
9.
 ⎧y < -x + 10
​ ​​      
    
​
 
1  ​x  + 7
_
y
<
​ 
⎩
10
⎨ 
8
⎧  y ≤ _
​  1 ​x  - 5
2
10. ​  ​​      
  
 ​
 ⎩ y ≥ -2x + 12
⎨ 
y
4
y
8
4
x
-8
-4
0
4
-4
x
8
-8
-4
-8
⎧  y ≤ -_
​  3 ​x 
5
    
11. ​  ​​
  
​
⎩  y > -x - 4
⎨ 
8
⎧  y ≥ 2x + 6
12. ​  ​​      
  
 ​
 
1 ​ x - 1
_
y
<
-​ 
⎩
2
⎨ 
y
4
0
4
-4
8
8
y
4
x
-8
-4
0
4
-4
x
-8
8
-4
-8
⎧  y ≤ _
​  4 ​x  - 4
5
​
13. ​  ​​       
 ⎩ y < 2x - 8
⎨ 
0
4
-4
8
-8
8
 ⎧x ≥ -6
   
14. ⎨ 
 ​
 
 ​ ​​ 
⎩y < 3
y
4
8
y
4
x
x
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-8
-4
0
-4
4
-8
8
-4
4
8
-8
-8
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Lesson 2
Graph each system of linear inequalities. Describe the solutions.
⎧  y ≤ 3x + 6
15. ⎨ 
​  ​​      
 
​
⎩ y < 3x - 8
8
y
⎨ 
4
x
-8
-4
0
4
-4
​  2 ​x  + 4
 ⎧y ≥ _
5
    
16. ​ ​​ 
 
 
​
 
2 ​ x - 6
_
y
≤
​ 
⎩
5
8
4
x
-8
8
-4
-8
​  5 ​x  - 6
 ⎧y ≥ _
4
 
 ​
17. ​ ​​     
 
5
_
⎩ y ≥ ​  4 ​ x 
⎨ 
8
y
0
4
-4
8
-8
​  3 ​x  - 3
 ⎧y ≥ -_
2
  
   ​
18. ​ ​​      
 
3
_
⎩ y ≤ -​  2 ​x  + 10
y
⎨ 
4
8
y
4
x
-8
-4
0
4
-4
x
-8
8
-4
-8
 ⎧x < 6
   
19. ⎨ 
​
  
 ​ ​​ 
⎩ x ≥ -3
8
0
4
-4
8
-8
​  9 ​x  - 1
 ⎧y ≥ _
4
​
20. ​ ​​      
 
9
_
⎩ y < ​  4 ​x  - 9
⎨ 
y
4
8
y
4
x
-4
0
-4
4
-8
8
-8
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Lesson 2
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-8
x
​  3 ​x  + 3
 ⎧y < -_
5
21. ​ ​​      
   ​
 
3
_
⎩ y ≥ -​  5 ​x  - 4
⎨ 
8
y
⎨ 
4
x
-8
-4
0
-4
_ 
1
 ⎧y > -​   ​x + 5
2
22. ​ ​​      
   ​
  y > -_
​  1 ​x  - 1
⎩
2
4
8
y
4
x
-8
8
-4
-8
0
4
-4
8
-8
H.O.T. Focus on Higher Order Thinking
23. Persevere in Problem Solving Write and graph a system of linear
inequalities for which the solutions are all the points in the second
quadrant, not including points on the axes.
8
y
4
x
-8
-4
0
-4
4
8
-8
24. Critical Thinking Can the solutions of a system of linear inequalities be the points on a line? Explain.
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y<_
​  32  ​x - 8
 ⎧   
​  ​​  
 
 ​ 
and
25. Explain the Error A student was asked to graph the system ⎨ 
3
__
y
≤
​ 
  ​x + 2
⎩
2
describe the solution set. The student gave the following
answer. Explain what the student did wrong, then give the correct answer.
The solutions are the same as the solutions of y ≤ __​  32 ​x  + 2.
8
y
8
4
y
4
x
-8
-4
0
-4
4
x
-8
8
-8
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Lesson 2
Lesson Performance Task
Successful stock market investors know a lot about inequalities. They know up to what point
they are willing to accept losses, and at what point they are willing to “lock in” their profits and
not subject their investments to additional risk. They often have these inequalities all mapped
out at the time they purchase a stock, so they can tell instantly if they are sticking to their
investment strategy. Graph the system of linear inequalities. Then describe the solution set and
give two ordered pairs that are solutions and two that are not. Is there anything particular to
note about the shape of this system?
⎧ 
3 ​ x + 4
y < -​ _
5
​ ​      
  
 ​
3 ​ x + 8
 y ≤ _
​ 
⎪       
2
​   ​​    ​
⎨ 
3 ​ x - 8
⎪   y > -​ _
5
 ​
  
​ ​ ​     
 y>_
​  3 ​ x - 6
2
⎩
⎪ 
⎪ 
8
y
4
x
-8
-4
0
-4
4
8
-8
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Lesson 2