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```6-8 Systems of Linear Inequalities
Select 2 colored pencils that blend together
to produce a third distinct color!
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
Systems of Linear Inequalities
Two or more linear inequalities in the same variables form a
system of linear inequalities.
A solution of a system of linear inequalities in two variables
is an ordered pair that is a solution of each inequality in
the system.
When graphing a system of linear inequalities, graph each
inequality in the same coordinate plane.
Do you remember how to graph an inequality?
Graphing A Linear Inequality
The graph of a linear inequality in
two variables is the graph of the
solutions of the inequality.
A boundary line divides the
coordinate plane into two
half-planes.
The solution of a linear inequality
in two variables is a half-plane.
Shade the side of the half-plane
that contains the solutions.
y
x
Graphing A Linear Inequality
The graph of a linear inequality in
two variables is the graph of the
solutions of the inequality.
A boundary line divides the
coordinate plane into two
half-planes.
The solution of a linear inequality
in two variables is a half-plane.
Shade the side of the half-plane
that contains the solutions.
y
x
Graphing A Linear Inequality
Use a dashed boundary line for < or >.
A dashed line indicates that the
points on the line are NOT
solutions.
Use a solid boundary line for < or >.
A solid line indicates that the
points on the line are solutions.
y
x
Graphing A Linear Inequality
1. Graph the corresponding equation, using a dashed
(< or >) line or a solid ( < or >) line. You may need to write
the equation in slope-intercept form to graph the
boundary line.
2. Test the coordinates of a point in one of the half-planes.
You can use any point that is not on the line as a test
point.
3. Shade the half-plane containing the test point if it is a
solution of the inequality. If it is not a solution, shade
the other half-plane.
Graph the inequality.
5x  3y  9
boundary line
5x  3y  9
 5x
 5x
3y  5x  9
3
3 3
5
y   x 3
3
Test (0,0)
5x  3y  9
50   30   9
09
false
y
•
•
•
x
Graph the system x < 1 and y > 2.
Write one inequality.
Write the related
equation.
Graph the equation;
solid boundary line.
Test a point. Use
(0,0) if possible.
x<1
x 1
x 1
0 1
true
y
•
x
Use the
y 2
inequality
y 2
half-plane that does
to test the
dashed
boundary line
The
overlap
It is not a
contain
thethe
test point.
point!
Would
Test (0,0)
represents
the
solution
ordered
pair
Test false
above
steps to
solution
of
the
because it
(1,2)
be
a
plane
that
does
not
graph the other
system. That
is
the
must be true
solution
of
contain of
thethe
test point.
inequality on the same
intersection
for
both
the system?
coordinate
plane.
two inequalities!
inequalities!
Graph each system.
Example 1 x > 2 and
y > –3.
Example 2 – x + y > – 5 and
Example 3 –x +2y > –2 and
4
y   x  2.
3
–x + 2y < 4.
Example 4 y < x + 1 and x > 1 and y > 2.
the above
problems in your
spiral notebook!
Example 1 Graph the system x > 2 and y > –3.
x>2
x 2
x2
02
false
y  3
y  3
dashed boundary line
0  3
true
Use the
inequality
to test the
point!
y
•
The solution is
the
intersection of
the two
inequalities!
x
4
3
Example 2 Graph the system – x + y > – 5 and y   x  2.
 x  y  5
 x  y  5
y  x 5
Test (0,0) in
original inequality.
half-plane that
contains test point.
4
y   x 2
3
4
y   x 2
3
Test (0,0) in
original inequality.
half-plane that
contains test point.
y
•
•
•
The solution is
the intersection
of the two
inequalities!
•
x
Example 3 Graph the system –x +2y > –2 and –x + 2y < 4.
 x  2y  4
 x  2y  2
 x  2y  4
 x  2y  2
2y  x  2
2y  x  4
1
1
y  x 1
y  x 2
2
2
Test (0,0) in
Test (0,0) in
original inequality. original inequality.
y
•
•
•
•
half-plane that
half-plane that
contains test point. contains test point. The solution is
the intersection
of the two
inequalities!
x
Example 4 Graph the system y < x + 1 and x > 1 and y > 2.
y  x 1
y  x 1
•
•
y
x
Example 4 Graph the system y < x + 1 and x > 1 and y > 2.
y  x 1
y  x 1
x 1
x 1
y 2
y 2
•
•
y
Note the region that
is the intersection of
these two
inequalities!
The intersection
of all 3
inequalities
represents the
solution of the
system.
x
6-A14 Pages 343-345 # 7–18, 37, 47-50.
```