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Solving and
Graphing
Inequalities
CHAPTER 6 REVIEW

An inequality is like an equation,
but instead of an equal sign (=) it
has one of these signs:
< : less than
≤ : less than or equal to
> : greater than
≥ : greater than or equal to

“x < 5”
means that whatever value x
has, it must be less than 5.
Try to name ten numbers that
are less than 5!

Numbers less than 5 are to the left
of 5 on the number line.
0 5 10 15
-20 -15 -10 -5
-25 20 25
• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.
• There are also numbers in between the integers, like
2.5, 1/2, -7.9, etc.
• The number 5 would not be a correct answer,
though, because 5 is not less than 5.

“x ≥ -2”
means that whatever value x
has, it must be greater than or
equal to -2.
Try to name ten numbers that
are greater than or equal to -2!

Numbers greater than -2 are to the
right of 5 on the number line.
0 5 10 15
-20 -15 -10 -5
-25 20 25
• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.
• There are also numbers in between the integers,
like -1/2, 0.2, 3.1, 5.5, etc.
• The number -2 would also be a correct answer,
because of the phrase, “or equal to”.
-2

Where is -1.5 on the number line?
Is it greater or less than -2?
0 5 10 15
-20 -15 -10 -5
-25 20 25
• -1.5 is between -1 and -2.
• -1 is to the right of -2.
• So -1.5 is also to the right of -2.
-2

Inequalities and their Graphs
5

x
5

x
7
6
3 5
4
2 8

Objective: To write and graph simple
inequalities with one variable

7
6
3 5
4
2 8
What is a good definition for Inequality?
An inequality is a statement that
two expressions are not equal

Terms you see and need to know to graph inequalities correctly
Notice
open
circles
< less than
> greater than

Terms you see and need to know to graph inequalities correctly
Notice colored in circles
≤ less than or equal to
≥ greater than or equal to

Let’s work a few together
3

x
3
Notice: when variable is on
left side, sign shows
direction of solution

7
7

x

-2
2


p
Color in
circle

8
Color in circle
8

x

Solve an Inequality
w + 5 < 8
w + 5 + (-5) < 8 + (-5)
w < 3
All numbers less
than 3 are
solutions to this
problem!

More Examples
8 + r ≥ -2
8 + r + (-8) ≥ -2 + (-8)
r ≥ -10
All numbers from -10 and up (including
-10) make this problem true!

More Examples
x - 2 > -2
x + (-2) + (2) > -2 + (2)
x > 0
All numbers greater than 0 make this
problem true!

More Examples
4 + y ≤ 1
4 + y + (-4) ≤ 1 + (-4)
y ≤ -3
All numbers from -3 down (including -3)
make this problem true!

There is one special case.
●Sometimes you may have to reverse the
direction of the inequality sign!!
●That only happens when you
multiply or divide both sides of the
inequality by a negative number.

Solving by multiplication of a
negative #
Multiply each side by the same negative number
and REVERSE the inequality symbol.
4

 x Multiply by (-1).
4


x
(-1) (-1)
See the switch

Solving by dividing by a negative
#
Divide each side by the same negative
number and reverse the inequality symbol.
6
2 
 x
3

x
-2 -2

Example:
Solve: -3y + 5 >23
-5 -5
-3y > 18
-3 -3
y < -6
●Subtract 5 from each side.
●Divide each side by negative 3.
●Reverse the inequality sign.
●Graph the solution.
0
-6

Try these:
1.) Solve 2x + 3 > x + 5 2.)Solve - c – 11 >23
3.) Solve 3(r - 2) < 2r + 4
-x -x
x + 3 > 5
-3 -3
x > 2
+ 11 + 11
-c > 34
-1 -1
c < -34
3r – 6 < 2r + 4
-2r -2r
r – 6 < 4
+6 +6 r < 10

You did remember to reverse
the signs . . .
5
7
4
15 



 x
7

7

12
4
8 


 x
4

7

4
 4

2 x 3

 
Good job!

Example: 8
4
6
2 

 x
x
- 4x - 4x
8
6
2 

 x
+ 6 +6
14
2 
 x
-2 -2
Ring the alarm!
We divided by a
negative!
7


x
We turned the sign!

Solving and Graphing Inequalities
Very Basics of Graphing Inequalities (on a number
line)
https://www.youtube.com/watch?v=nif2PKA9bXA
Graphing an inequality with the variable on the
right side and negative
https://www.youtube.com/watch?v=Em_Taf3_aRo

Ex: Solve 6x-3 = 15
6x-3 = 15 or 6x-3 = -15
6x = 18 or 6x = -12
x = 3 or x = -2
* Plug in answers to check your solutions!

Ex: Solve 2x + 7 -3 = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.

Ex: Solve & graph.
• Becomes an “and” problem
21
9
4 

x
2
15
3 

 x -3 7 8

Solve & graph.
• Get absolute value by itself first.
• Becomes an “or” problem
11
3
2
3 


x
8
2
3 

x
8
2
3
or
8
2
3 



 x
x
6
3
or
10
3 

 x
x
2
or
3
10


 x
x
-2 3 4

Example 1:
● |2x + 1| > 7
● 2x + 1 > 7 or 2x + 1 >7
● 2x + 1 >7 or 2x + 1 <-7
● x > 3 or x < -4
This is an ‘or’ statement.
(Greator). Rewrite.
In the 2nd
inequality, reverse the
inequality sign and negate the
right side value.
Solve each inequality.
Graph the solution.
3
-4

Example 2:
● |x -5|< 3
● x -5< 3 and x -5< 3
● x -5< 3 and x -5> -3
● x < 8 and x > 2
● 2 < x < 8
This is an ‘and’ statement.
(Less thand).
Rewrite.
In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
Solve each inequality.
Graph the solution.
8
2

Absolute Value Inequalities
Case 1 Example: 3 5
x  
3 5
2
x
x
  
 
and
8
x
5
3
x



8
x
2 



Absolute Value Inequalities
Case 2 Example: 2 1 9
x  
2 1 9
2 10
5
x
x
x
  
 
 
5
x  
2 1 9
2 8
4
x
x
x
 


4
x 
OR
or

Absolute Value
• Answer is always positive
• Therefore the following examples
cannot happen. . .
Solutions: No solution
9
5
3x 



Graphing Linear Inequalities
in Two Variables
•SWBAT graph a linear
inequality in two variables
•SWBAT Model a real life
situation with a linear
inequality.

Some Helpful Hints
•If the sign is > or < the line is
dashed
•If the sign is  or  the line will
be solid
When dealing with just x and y.
•If the sign > or  the shading
either goes up or to the right
•If the sign is < or  the shading
either goes down or to the left

When dealing with slanted lines
•If it is > or  then you shade above
•If it is < or  then you shade below
the line

Graphing an Inequality in Two Variables
Graph x < 2
Step 1: Start by graphing
the line x = 2
Now what points
would give you less
than 2?
Since it has to be x < 2
we shade everything to
the left of the line.

Graphing a Linear Inequality
Sketch a graph of y  3

Using What We Know
Sketch a graph of x + y < 3
Step 1: Put into
slope intercept form
y <-x + 3
Step 2: Graph the
line y = -x + 3

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