Graphing Linear Inequalities
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Graphing Linear Inequalities in Slope-Intercept Form
Graphing Linear Inequalities NOT in Slope-Intercept Form
Graphing Linear Inequalities
in
Slope-Intercept Form
(y = mx +b)
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Table of Contents
Graphing
Graphs of inequalities are similar to linear equations because they both have
points on a coordinate plane and a line connecting the points. However, a linear
equation is ONLY the line but an inequality extends beyond that line.
Linear Equation: y = 2x + 1
Inequality: y < 2x-1
The following are graphs of linear inequalities:
y > mx + b
y > mx + b
y < mx + b
How do the graphs
at left
compare with the
graph below for
y = mx + b?
y < mx + b
Next slide for
observations
The following are graphs of linear inequalities:
y > mx + b
Shading is above a dotted line.
This means the answers are
above the line but NOT on it.
y > mx + b
Shading is above a solid line.
This means the answers are
above the line AND on it.
y < mx + b
Shading is below a dotted line.
This means the answers are
below the line but NOT on it.
y < mx + b
Shading is below a solid line.
This means the answers are
below the line AND on it.
How do the graphs
at left
compare with the
graph below for
y = mx + b?
How to Graph a Linear Inequality
Think
y = mx + b
to graph the boundary
1) Decide where the boundary goes:
Solve inequality for y, for example y > 2x - 1
2) Decide whether boundary should be:
solid (< or >: points on the boundary make the inequality true) or
dashed (< or >: points on the boundary make the inequality false)
3) Graph the boundary (the line)
4) Decide where to shade:
y > or y >: shade above (referring to y-axis) the boundary
y < or y <: shade below (referring to y-axis) the boundary
(Or, you can test a point, which will be explained later)
EXAMPLE 1
Graph y < -2x + 1
Step 1: Solve for y: Think y = -2x + 1, m = -2 and b = 1
Step 2: The line should be dashed because the inequality is <
Step 3: Graph boundary
Step 4: Shade below the boundary line because y <
Example 2
Graph 2x - y < 4
Step 1: Solve for y
-y < -2x + 4
y > 2x - 4
Step 2: The line should be solid because the inequality is >
Step 3: Graph boundary
Step 4: Shade above the boundary line because y >
Example 3
Graph
Step 1: Solve for y
Step 2: The line should be dashed because the inequality is >
Step 3: Graph boundary
Step 4: Shade above the boundary line because y >
Why are there dashed boundaries on some graphs of inequalities?
A
Points on the line make the inequality false.
B
Points on the line make the inequality true.
C
The slope of the line depends on the line type.
D
The y-intercept depends on the line type.
answer
1
For which of these equations would the graph
have a solid boundary and be shaded above?
A
y < 3x-2
B
y < 3x-2
C
y > 3x-2
D
y > 3x-2
answer
2
3
A
y < 3x-2
B
y < 3x-2
C
y > 3x-2
D
y > 3x-2
answer
For which of these equations would the graph
have a dashed boundary and be shaded above?
A
y < 3x-2
B
y < 3x-2
C
y > 3x-2
D
y > 3x-2
answer
Which inequality is
graphed?
4
A
y < 3x-2
B
y < 3x-2
C
y > 3x-2
D
y > 3x-2
answer
Which inequality is
graphed?
5
Which inequality could match the given graph?
A
y>3
B
y<3
C
x<3
D
x>3
answer
6
Graphing Linear Inequalities
NOT in Slope-Intercept Form
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Table of Contents
Inequalities can be graphed without converting to
slope-intercept form.
The steps to graph are the same but determining where to shade involves a
different step.
1) Decide where the boundary goes:
2) Decide whether boundary should be:
solid (< or >: points on the boundary make the inequality true) or
dashed (< or >: points on the boundary make the inequality false)
3) Graph the boundary (the line)
4) Decide where to shade:
Choose a test point on the graph and shade accordingly
Test Points
The shaded region represents all of the points that make the inequality true, so:
A) Select a point NOT on the boundary and substitute into
the inequality
(The point (0, 0) is an easy value to work with if it is not on the boundary)
B) If
your selected test point makes the inequality TRUE,
shade the region containing your test point
C) If your selected test point makes the inequality FALSE,
shade the region opposite your test point
Example 4
Graph y > -4x
Step 1: Solve for y
y > -4x
Step 2: The line should be dashed because the inequality is >
Step 3: Graph boundary
Step 4: Test a point
-Choose a point and substitute it into the inequality
(0,-5):
y > -4x
-5 > -4(0)
-5 > 0
Statement is FALSE so the opposite region will be shaded
o
(0,-5)
Example 5
Graph y < 2x + 5
Step 1: Solve for y
y < 2x + 5
Step 2: The line should be solid because the inequality is <
Step 3: Graph boundary
Step 4: Test a point
-(0, 0) is an easy point to work with and is not on the
boundary
Example 5 Continued
y < 2x + 5
Substitute (0, 0) into the inequality
0 < 2(0) + 5
0<5
o
(0,0)
This statement is TRUE so shade the region of the graph where this point occurs
Example 6
Graph y - 2 < -2(x + 1)
Step 1: Solve for y
y - 2 < -2(x + 1)
y - 2 < -2x - 2
y < -2x
Step 2: The line should be dashed because the inequality is <
Step 3: Graph boundary
Step 4: Test a point
-(0, 0) is on the graph so it cannot be used but try to pick
a point with a 0 in it to make the substitution easier
Example 6 Continued
Example test point:
o
(5,0)
Substitute the test point into the inequality:
y < -2x
0 < -2(5)
o
(5,0) FALSE
0 < -10
The test point produces a FALSE statement, so we shade the opposite
region.
Note: You always know with certainty which side to shade when you use
test points.
What point can be used as test points when graphing ANY
inequality?
A
Only the origin.
B
Any point with a zero in the ordered pair.
C
Any point.
D
Any point not on the boundary.
answer
7
8
Given the inequality,
, which test point
should be used and where should the graph be
shaded?
A
(0,0): shade above boundary
(0,4)
(0,0); shade below boundary
C
(0,4): shade above boundary
D
(0,4); shade below boundary
(0,0)
answer
B
9
Given the inequality,
test
point should be used and where is the shading?
A
, which
(0, 0), shade below boundary
(-4,2)
B
(-4,2), shade below boundary
C
(0, 0), shade above boundary
D
(-4,2), shade above boundary
answer
(0,0)
Given the inequality 6x + 10y > 30, which is the easiest test point to
use and where should the graph be shaded?
A
(5, 3), shaded below
B
(0, 0), shaded below
C
(5, 3), shaded above
D
(0, 0), shaded above
answer
10
11
Given the inequality,
region?
B
answer
A
, which is the graph of the solution
C
D