Math 95
Section 9.4
Absolute Value Inequality
Objective:
Students will be able to
 Solve absolute value inequalities
Since we a working with inequalities, here are some things that are important to
remember when setting up absolute inequalities.
Inequality
x 1
Connecting Word
AND
x 1
AND
x 1
OR
Setup
1  x  1
or
x  1 and x  1
1  x  1
or
x  1 and x  1
x  1 or x  1
x 1
OR
x  1 or x  1
This chart should help you when you are setting up the problems.
Example:
Solve: y  2  0 . Graph the solution on a number line and write the solution set in
interval notation.
So just like with the absolute value equations, we have to make sure that the absolute
value is by itself. Since the problem is already like that, we are ready to start.
y20
y22  02
or
or
y20
y22  02
y  2
or
y  2
][ ‐2 Since we are working with “or”, we know that the solution must come from one, the
other, or both. Well by looking at the number line, you should notice that everything is
included. So your solution is the whole number line.
Interval Notation:  ,  
Example:
Solve: 2 x  7  4  5 . Graph the solution on a number line and write the solution set in
interval notation.
So just like with the absolute value equations, we have to make sure that the absolute
value is by itself.
2x  7  4  4  5  4
2x  7  1
Now we are ready to solve.
1  2 x  7  1
1  7  2 x  7  7  1  7
6  2x  8
6 2x 8


2 2 2
3 x  4
( ) 3
4
Interval Notation:  3, 4 
Example:
Solve: k  7  3 . Graph the solution on a number line and write the solution set in
interval notation.
So just like with the absolute value equations, we have to make sure that the absolute
value is by itself. Since the problem is already like that, we are ready to start.
k  7  3
and
k 7 3
k  7  7  3  7
and
k 7  7  3 7
k4
and
k  10
) ( 4 10 Since we are working with “and”, we know that in order to have a solution that there has
to be an overlap over values on the number line. Well when we look at the number line,
we notice that there is no overlapping. So what this tells us is that there is no solution.
So the number line will look like this:
Interval Notation:

Example:
Solve: 7 y  1  3  11 . Graph the solution on a number line and write the solution set in
interval notation.
So just like with the absolute value equations, we have to make sure that the absolute
value is by itself.
7 y  1  3  3  11  3
7 y  1  14
7
14
y 1 
7
7
y 1  2
Now we are ready to solve.
y 1  2
or
y  1  2
y 1 1  2 1
y 1
or
or
y  1  1  2  1
y  3
] [ ‐3 1 Since we are working with “or”, we know that the solution must come from one, the
other, or both. Well by looking at the number line, you should notice that we don’t have
any overlapping. So there will be no “both” to consider in the solution.
Interval Notation:  , 3  1,  