Warm-Up
Graph each inequality on a number line.
1. x > –5
2. y ≤ 0
3. Write –6x + 2y = –4
in slope-intercept form,
and graph.
y = 3x – 2
Inequalities
Learning Targets
 Review Inequalities
 Graphing Inequalities
 Knowing where Possible Solutions
Exist
 Testing for Possible Solutions
What is an Inequality?
 Definition: two expressions are not
equal to one another.
Equality vs. Inequality
 Equality sets expressions values so that they
are equal to one another.
 Ex: 4 = 4
 Inequality compares the two expressions
values.
 Ex: 3 < 4
Number Lines
 Equalities on a number
line:
-1
0
1
-1
0
1
-1
0
1
-1
0
1
 Inequalities on a
number line:
Graphically
 Equalities on a
coordinate plane:
 Inequalities on a
coordinate plane:
𝑦 = 2𝑥 + 3
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
𝑦 ≤ 2𝑥 + 3
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
Different Symbols…
Different Lines/Points
≤, ≥
<, >
Solid Line,
Filled Point
Dashed Line,
Hollow Point
Objective - To graph linear inequalities in
the coordinate plane.
Number Line
x3
Coordinate Plane
x3
y
-4 -3 -2 -1 0 1 2 3 4
x
x=3
Graph y  2.
Number Line
Coordinate Plane
y  2
y  2
-4 -3 -2 -1 0 1 2 3 4
y
x
y = -2
2
Graph y  x  1.
3
y
Boundary Line
2
y  x 1
3
2
m
b  1
3
Test a Point
2
y  x 1
3
2
0  (0)  1
3
0  1 False!
x
Graph 4x  5y  10.
4x  5y  10
4x
 4x
5y  4x  10
5
5

4
y
x2
5

4
4
m

5
5
b2
If y = mx + b,

Solid line
y
x
Graph 3x  2y  8.
3x  2y  8
3x
 3x
2y  3x  8
2
2
Why do we
flip the sign?
Pause and
Ponder…
Graph 3x  2y  8.
3x  2y  8
3x
 3x
2y  3x  8
2
2
3
y x4
2
3

3
m 
2 2
b  4
If y = mx + b,

Dashed line
y
x
Graph 4x  3y  6.
4x  3y  6
4x
 4x
3y 
 4x  6
3
3
4
y x2
3
4

4
m 
3 3
b  2
If y = mx + b,

Solid line
y
x
Write the inequality described by the graph below.
b2
y
4
m
3
If y = mx + b,
Dashed line
4
y x2
3

-4
+3
x
Determine whether the given point is a solution
to the inequality -2x + 3y < 9.
(x, y)
1) (2, -3)
2x  3y  9
2(2)  3(3)  9
4  9  9
13  9
Yes, (2,-3) is a solution.
2) (3, 5)
2x  3y  9
2(3)  3(5)  9
6  15  9
99
No, (3,5) is
not a solution.
Problem
If you have less than $5.00 in nickels and dimes,
find an inequality and sketch a graph to describe
how many of each coin you have.
Let n = # of nickels
Let d = # of dimes
0.05 n + 0.10 d
< 5.00
or
5 n + 10 d < 500
5n + 10d < 500
n
d
d
0
50
60
100
0
50
40
30
20
10
0
n
0 10 20 30 40 50 60 70 80 90 100
𝑦 < −3𝑥 + 10
11
10
9
8
7
6
5
4
3
2
1
-11 -10 -9 -8
-7 -6
-5
-4 -3
-2
-1 -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
𝑦 ≤𝑥+1
y
11
10
9
8
7
6
5
4
3
2
1
x
1
2
3
4
5
6
7
8
9
10 11
-11 -10 -9 -8
-7 -6
-5
-4 -3
-2
-1 -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
y
x
1
2
3
4
5
6
7
8
9
10 11
Graph both
inequalities over
each other:
Therefore our
answer lies in the
combined shaded
region:
11
10
9
8
7
6
5
4
3
2
1
-11 -10 -9 -8
-7 -6
-5
-4 -3
-2
y
x
-1 -1
1
2
3
4
5
6
7
8
9
10 11
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
11
10
9
8
7
6
5
4
3
2
1
-11 -10 -9 -8
-7 -6
-5
-4 -3
-2
-1 -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
y
x
1
2
3
4
5
6
7
8
9
10 11
Non linear Inequalities
Non Linear Inequalities
 In chapter 4 we studied parent graphs and their
transformation.
 In the next few slides you will be given non linear
graphs. Use your knowledge of parent graphs to
display a graph for each function.
 Then use a test point to graph the inequalities that
follow.
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1 2 3 4 5 6 7 8 9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
On your own:
 Review your
notes. Rewrite
and fortify them
if needed.
 Update your
vocab list, if
needed.