ALEKS PURPLE Solving an inequality involving absolute value
Each time you deal with absolute value, remember what absolute value means:
Whatever is inside the absolute value symbol becomes positive.
If you are asked to solve an inequality involving absolute value (with the variable inside
the absolute value symbol), you will end up with two equations and generally two
solutions. Let’s look at an example:
2x  6  10
You are told that the expression is less than 10, but because the absolute value symbol
forces it to be positive, the expression MAY have been less than -10 without the
absolute value symbol. Thus, you have TWO equations to solve:
2x  6  10
and
2x  6  10
How did I get those two inequalities? Remember the following rules for absolute value
inequalities:
A  B becomes A  B and A  B
A  B becomes A  B or
A  B
If you can’t remember this rule, put numbers in the place of A and B:
4  5 becomes
4?5
and
4 ?  5  and because they overlap
4  5 becomes
4?5
or
4 ?  5  or because no overlapping
I am sure by this point that you could solve those two inequalities, so let’s look at a
more complex inequality involving absolute value:
19  9 z  1  8
In this example, the absolute value expression is not all by itself on one side of the
inequality. In order to solve this inequality, you must have the absolute value
expression by itself. So we will “move” the 8 to the other side and divide by 9 to
“move” that to the other side:
19  9 z  1  8
Add 8 to both sides:
19  8  9 z  1  8  8
19  8  9 z  1 8 8
27  9 z  1
Divide both sides by 9:
9z 1
27

9
9
3
9 z 1
27

9
9
3  z 1
We can rewrite this inequality as: z  1  3
Now that we have the absolute value by itself, we can write it as TWO inequalities and
eliminate the absolute value symbol:
z 1  3
AND
z  1  3
To solve these, you will want to subtract 1 from both sides of the inequality. You’ll end
up with the final answers:
z 2
and
z  4
Because these overlap, you could also write it:
4  z  2