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Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Systems of Linear Inequalities
Solving Linear Inequalities in Two Variables
Solving Systems of Linear Inequalities
4.3

When the equals sign in a linear equation in two
variables is replaced with one of the symbols <, ≤, >,
or ≥, a linear inequality in two variables results.
Examples:
x > 4 y ≥ 2x – 3
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Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1
6
2
x y
 

�ݺ�ߣ 3
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Solving linear inequalities
Shade the solution set for each inequality.
a. b. c.
a. Begin by graphing a vertical line x = 3 with a dashed
line because the equality is not included.
3
x 
The solution set includes all
points with x-values greater
than 3, so shade the region
to the right of the line.
3 2
y x
  3 6
x y
 

�ݺ�ߣ 4
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Solving linear inequalities-continued
Shade the solution set for each inequality.
a. b. c.
b. Begin by graphing the line y = 3x – 2 with a solid line
because the equality is included.
3
x 
Check a test point.
Try (0, 0)
0 ≤ 3(0) – 2
0 ≤ – 2
False (shade the side NOT
containing (0, 0).
3 2
y x
  3 6
x y
 

�ݺ�ߣ 5
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Solving linear inequalities-continued
Shade the solution set for each inequality.
a. b. c.
c. Begin by graphing the line. Use intercepts or slope-
intercept form. The line is dashed.
3
x 
Check a test point.
Try (0, 0)
0 – 3y < 6
0 – 0 < 6
0 < 6
True (shade the side
containing (0, 0).
3 2
y x
  3 6
x y
 

�ݺ�ߣ 6
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solving Systems of Linear Inequalities
A system of linear inequalities results when the
equals sign in a system of linear equations are replaced
with <, ≤, >, or ≥.
The solution to a system of inequalities must satisfy
both inequalities.
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Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

�ݺ�ߣ 8
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Solving a system of linear inequalities
Shade the solution set for each system of inequalities.
a. b. c.
a. Graph each line as a solid line.
1
3
x
y


Shade each region.
Where the regions overlap is
the solution set.
3
4
y x
x y

 
3 4
2 8
x y
x y
 
 

�ݺ�ߣ 9
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Solving a system of linear inequalities
Shade the solution set for each system of inequalities.
a. b. c.
b. Graph each line as a solid line.
1
3
x
y


Shade each region.
Where the regions overlap is
the solution set.
3
4
y x
x y

 
3 4
2 8
x y
x y
 
 

�ݺ�ߣ 10
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Solving a system of linear inequalities
Shade the solution set for each system of inequalities.
a. b. c.
c. Graph each line < is dashed and ≥ is solid.
1
3
x
y


Shade each region.
Where the regions overlap is
the solution set.
3
4
y x
x y

 
3 4
2 8
x y
x y
 
 

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linineq2.pptlinineq2.pptlinineq2.pptlinineq2.ppt

2. When the equals sign in a linear equation in two variables is replaced with one of the symbols <, ≤, >, or ≥, a linear inequality in two variables results. Examples: x > 4 y ≥ 2x – 3 �ݺ�ߣ 2 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 6 2 x y  

3. �ݺ�ߣ 3 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Solving linear inequalities Shade the solution set for each inequality. a. b. c. a. Begin by graphing a vertical line x = 3 with a dashed line because the equality is not included. 3 x  The solution set includes all points with x-values greater than 3, so shade the region to the right of the line. 3 2 y x   3 6 x y  

4. �ݺ�ߣ 4 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Solving linear inequalities-continued Shade the solution set for each inequality. a. b. c. b. Begin by graphing the line y = 3x – 2 with a solid line because the equality is included. 3 x  Check a test point. Try (0, 0) 0 ≤ 3(0) – 2 0 ≤ – 2 False (shade the side NOT containing (0, 0). 3 2 y x   3 6 x y  

5. �ݺ�ߣ 5 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Solving linear inequalities-continued Shade the solution set for each inequality. a. b. c. c. Begin by graphing the line. Use intercepts or slope- intercept form. The line is dashed. 3 x  Check a test point. Try (0, 0) 0 – 3y < 6 0 – 0 < 6 0 < 6 True (shade the side containing (0, 0). 3 2 y x   3 6 x y  

7. Solving Systems of Linear Inequalities A system of linear inequalities results when the equals sign in a system of linear equations are replaced with <, ≤, >, or ≥. The solution to a system of inequalities must satisfy both inequalities. �ݺ�ߣ 7 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

8. �ݺ�ߣ 8 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Solving a system of linear inequalities Shade the solution set for each system of inequalities. a. b. c. a. Graph each line as a solid line. 1 3 x y   Shade each region. Where the regions overlap is the solution set. 3 4 y x x y    3 4 2 8 x y x y    

9. �ݺ�ߣ 9 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Solving a system of linear inequalities Shade the solution set for each system of inequalities. a. b. c. b. Graph each line as a solid line. 1 3 x y   Shade each region. Where the regions overlap is the solution set. 3 4 y x x y    3 4 2 8 x y x y    

10. �ݺ�ߣ 10 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Solving a system of linear inequalities Shade the solution set for each system of inequalities. a. b. c. c. Graph each line < is dashed and ≥ is solid. 1 3 x y   Shade each region. Where the regions overlap is the solution set. 3 4 y x x y    3 4 2 8 x y x y    

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