Lesson 7.5, page 755
Systems of Inequalities
Objective: To graph linear inequalities,
systems of inequalities, and solve linear
programming problems.
Review -- Graphing a Line
1.
2.
3.
4.
Put in y = mx + b form.
Plot the y-intercept.
Use the slope and rise/run to plot at least 2
more points.
Draw the line.
Review
Practice Graphing Lines

Graph y = 3x + 5.
10

y
Graph 2x + 3y = 6.
10
y
x
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10
x
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10
Review
Linear Inequality in Two Variables



A linear inequality divides the xy-plane into 2 parts.
Either the points on one side of the line make the
inequality true or points on the other side do.
Once the line is determined, select any point on
either side to test in the original inequality to
determine if that point is a solution or not.
If the point makes the inequality true, all points on
that side are also in the solution set. If the point
makes the inequality false, all points on the other
side of the line are in the solution set.
Steps for Graphing Inequalities
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


FOR INEQUALITIES:
< or > dashed line
> or < solid line
Write the inequality as an equation. Put in
slope-intercept form and graph the line, dashed
or solid.
Test a point, not on the line to see if it makes a
true statement. If (0, 0) is not on the line, use it.
It is the easiest point to test.
If true, shade on the side of the line that contains
the test point.
If false, shade on the side of the line that does
not contain the test point.
Example

Graph y > x  4.

We begin by graphing the
related equation y = x  4. We
use a dashed line because the
inequality symbol is >. This
indicates that the points on the
line itself are not in the solution
set.


Test point (0,0)
y>x4
0?04
0 > 4 True
y
10
x
Determine which half-plane
satisfies the inequality and
shade.
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Example

Graph: 4x + 2y > 8

1. Graph the related equation,
using a solid line.

2. Determine which half-plane
to shade.
4x + 2y > 8
4(0) + 2(0) >? 8
0 > 8 is false.
We shade the region not
containing (0, 0).
y
10
x
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10
Check Points 1 & 2

4x - 2y > 8
10

y
10
y > -3x
4
y
x
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10
x
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10
Example 3, page 758
Check Point 3

y>1
10

y
10
x < -2
y
x
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10
x
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10
To Graph a Linear Inequality:
A Recap

Graph the related equation. If the inequality symbol
is < or >, draw the line dashed. If the inequality
symbol is  or , draw the line solid.

Because an inequality has many possible solutions,
the graph consists of a half-plane on one side of the
line and, if the line is solid, the line as well. To
determine which half-plane to shade, test a point
not on the line in the original inequality. If that point
is a solution, shade the half-plane containing that
point. If not, shade the opposite half-plane.
Nonlinear Inequalities
Consider the graph of a circle. The plan is
divided into the area inside the circle, and
that outside the circle. Solve it as you
would a linear system.
 Consider the graph of other nonlinear
inequalities (parabolas, ellipses,
hyperbolas). Again, the graph would show
that the points that make the inequality
true would be found inside OR outside of
the graph.

Graphing a Non-Linear Inequality
See Example 4, page 759

Check Point 4
x2 + y2 > 16
y
10
x
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System of Linear Inequalities
If the 2 lines intersect at one point, the
plane is divided into 4 areas. The solution
could be found in one of these areas.
 Often graphing and looking for overlapping
areas is easier than looking at points in
each region.

Steps for Graphing a
System of Inequalities
1.
2.
3.
Graph each inequality and indicate which
part should be shaded.
Shade the area which is common to all
graphs or the area where the shading
overlaps.
Pick any point in the commonly shaded
area and check it in all inequalities.
Systems of Linear Inequalities

Graph the solution set of the
system.
x y3
x  y 1

First, we graph x + y  3 using
a solid line.
Choose a test point (0, 0)
and shade the correct plane.

Next, we graph x  y > 1 using
a dashed line.
Choose a test point and
shade the correct plane.
The solution set of the system of
equations is the region shaded
both red and green, including
part of the line x + y  3.
Check Point 5, page 759
y
x – 3y < 6
 2x + 3y > -6
10

x
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Check Point 6, page 761
y > x2 – 4
x+y<2

y
10
x
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10
See Example 7, page 762.

Check Point 7: Graph the solution set of the system:
x y2
2  x  1
y
10
y  3
x
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