Transcript 3.3 Graph System of Linear Inequalities

Lesson 3.3
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.
Warm - Up
Graphing Lines
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
Graph y = 3x + 5.
10
y
Graph 2x + 3y = 6.
10
y
x
-10
-10
10
x
-10
-10
10
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.
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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.
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
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.
-10
-10
10
Example
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Graph: 4x + 2y > 8
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1. Graph the related equation,
using a solid line.
y
10
x
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2. Determine which half-plane
to shade.
4x + 2y > 8
4(0) + 2(0) >? 8
-10
-10
0 > 8 is false.
We shade the region not
containing (0, 0).
10
Example
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y>1
10
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y
10
x < -2
y
x
-10
-10
10
x
-10
-10
10
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.
Graphing Method
•
•
•
Example: Graph the
inequalities on the
same plane: x + y < 6
and 2x - y > 4.
Before we graph them
simultaneously, let’s
look at them separately.
Graph of x + y < 6. --->
Graphing Method
Now lets graph :
2x - y > 4.
Coolness Discovered!
•
•
Wow!
The solution to the
system is the
hashtag region where the two
shaded areas
coincide.
So what were the steps?
•
Graph first inequality
•
•
Graph second inequality
•
•
•
Shade lightly (or use colored pencils)
Shade lightly (or use colored pencils)
Shade darkly over the common region of
intersection.
That is your solution!
Systems of Linear Inequalities

Graph the solution set of the
system.
x y3
x  y 1
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First, we graph x + y  3 using
a solid line.
Choose a test point (0, 0)
and shade the correct plane.
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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.
Practice Problem
y
10
4x+ 2y > 8
 y < -2x-3
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x
-10
-10
10
Practice Problem
y>x–4
x+y<2
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y
10
x
-10
-10
10
What about THREE
inequalities?
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Graph x ≥ 0, y ≥ 0, and 4x + 3y ≤ 24
First off, let’s look at x ≥ 0 and y ≥ 0
y
y
10
10
separately.
x
-10
10
-10
x
-10
10
-10
Graphing THREE inequalities
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Now let’s look at x ≥ 0 and y ≥ 0 together.
10
y
Clearly, the solution
set is the first
quadrant.
x
-10
10
-10
Graphing THREE inequalities
•
So therefore, after we graph the third
inequality, we know the solution region will be
y
trapped inside the first quadrant. 10So
let’s look
at 4x + 3y ≤ 24 by itself.
x
-10
10
-10
Graphing THREE inequalities
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Now we can put all of our knowledge
together.
The solution region is the right triangle in the
y
10
first quadrant.
x
-10
10
-10
Homework
 Section
#1-3,
3.3 - pg.171-173
8-12, 16, 21, 25, 29, 31,
34, 35, 36