Download system of linear inequalities

4.3
Systems of Linear Inequalities
Solving Linear Inequalities in Two Variables
Solving Systems of Linear Inequalities
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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x y 6
2
Slide 2
EXAMPLE
Solving linear inequalities
Shade the solution set for each inequality.
a. x  3
b. y  3 x  2
c. x  3 y  6
Solution
a. Begin by graphing a vertical line x = 3 with a dashed
line because the equality is not included.
The solution set includes all
points with x-values greater
than 3, so shade the region
to the right of the line.
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3
EXAMPLE
Solving linear inequalities-continued
Shade the solution set for each inequality.
a. x  3
b. y  3 x  2
c. x  3 y  6
Solution
b. Begin by graphing the line y = 3x – 2 with a solid line
because the equality is included.
Check a test point.
Try (0, 0)
0 ≤ 3(0) – 2
0≤–2
False (shade the side NOT
containing (0, 0).
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4
EXAMPLE
Solving linear inequalities-continued
Shade the solution set for each inequality.
a. x  3
b. y  3 x  2
c. x  3 y  6
Solution
c. Begin by graphing the line. Use intercepts or slopeintercept form. The line is dashed.
Check a test point.
Try (0, 0)
0 – 3y < 6
0–0<6
0<6
True (shade the side
containing (0, 0).
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6
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.
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7
EXAMPLE
Solving a system of linear inequalities
Shade the solution set for each system of inequalities.
y  3x
a. x  1
b.
c. 3 x  y  4
y3
x y 4
x  2y  8
Solution
a. Graph each line as a solid line.
Shade each region.
Where the regions overlap is
the solution set.
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8
EXAMPLE
Solving a system of linear inequalities
Shade the solution set for each system of inequalities.
y  3x
a. x  1
b.
c. 3 x  y  4
y3
x y 4
x  2y  8
Solution
b. Graph each line as a solid line.
Shade each region.
Where the regions overlap is
the solution set.
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9
EXAMPLE
Solving a system of linear inequalities
Shade the solution set for each system of inequalities.
y  3x
a. x  1
b.
c. 3 x  y  4
y3
x y 4
x  2y  8
Solution
c. Graph each line < is dashed and ≥ is solid.
Shade each region.
Where the regions overlap is
the solution set.
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 10