Solving Absolute Value Equations
Solving absolute value equations is almost the exact same as solving regular equations with one
major difference. In most cases you have 2 solutions.
Example:
|x|=5
We know that when x = 5, | 5 | will also equal 5, but it is also true that | -5 | will equal 5. So, for
|x | = 5, x = {-5, 5}. They both work.
How to solve absolute value equations
1) Isolate the absolute value.
2) Split into two separate equations, setting one to the negative and one to the positive.
Example:
| 2x + 6 | - 3 = 13
1) Isolate the absolute value:
** The steps are the same as if you were getting the x by itself. You move away all other
numbers by doing the opposite operation:**
| 2x + 6 | - 3 = 13
+3 +3
| 2x + 6 |
= 16
2) Now split into two
separate equations and
solve each.
2x + 6 = -16
-6 -6
2x
= -22
2
2
2x + 6 = 16
-6 -6
2x
= 10
2
2
x = -11
x=5
3) Check by substituting in the original equation.
Do in NB (with checks):
1) | 6x + 12 | = 24
2) | 6 – 2x | = 14
3) | 8x – 2 | = 42
4) | ⅔ x + 6 | = 2
5) | 10 – ¾ x | = 16
6) | 4x – 12 | = -36
1
On #’s 1-10, notice how the steps of isolating the absolute value are the same as if you were
isolating the x.
1) 5x + 9 = 144
2) 5| 3x – 6 | + 9 = 144
3) x . – 3 = 1
7
4)
5) ⅔ x – 11 = -3
6) ⅔ |2x – 10| - 11 = -3
7) 4x – 5 .
3
8) 4| 8x – 16 | - 5 .
3
=9
2
| 12x – 8 | .
7
–3=1
=9
9) 5x + 7 .
11
–8=-6
10) 5| 6x -15 | + 7 .
11
– 8 = -6
11) | 4x – 5| + 15 = 36
12) 6| 3x – 12 | - 5 = 49
13) ⅝ |2x – 4| + 4 = -9
14) 4| 8x – 16 | - 5 .
7
3
=9
Absolute Value Inequalities
Solving absolute value inequalities combine the strategies you used in:
1) Solving and Graphing Compound Inequalities
2) Solving Absolute Value Equations
Every absolute value inequality is a compound inequality. The 2 separate inequalities come
from when you split the inequality once the absolute value is isolated.
So first, you isolate the absolute value following all the same steps as you did when isolating the
absolute value when solving an absolute value equation. Then you split and set to the negative
and the positive.
The new stuff:
1) Once the absolute value is isolated, you choose whether it will be an AND problem or an OR
problem.
- Greater than (> , > ) will be an OR problem. (Graph both and keep)
- Less than (<, < ) will be an AND problem. (Graph both and keep the intersection)
2) When you split the absolute value YOU MUST TURN THE INEQUALITY AROUND
WHEN SETTING TO THE NEGATIVE.
Example:
| 2x – 3 | - 10 > -5
+10 +10
Isolate the abs. val.:
| 2x – 3 |
Split and turn the ineq.
around (to less than)
when setting to -5 and
solve each ineq:
Since its a Greater than,
it is an OR problem and
will be graphed accordingly.
> 5
2x – 3 < -5
+3 +3
2x
< -2
2
2
x<-1
2x – 3 > 5
+3 +3
2x
>8
2
2
OR
x>4
<-------------------------------------------------------------------------------------->
4
Special Situations:
Just like when you were learning absolute value equations, if you isolate the absolute value and it
is greater than or less than a negative number, it is a special situation:
a) |4x + 2| = -10
b) |4x + 2| < -10
c) |4x + 2| > -10
Solve each inequality and graph in your notebook:
1) |5x| > 30
2) |-7x| < 14
3) |-3x| < -21
4) |5x - 10| > 15
5) 4| 3x – 6 | - 9 > 27
6) -6|4x – 12| - 11 > -131
7) ¾|8x – 16| + 11 < 17
8) ½ |5x + 10| - 22 > -12
9)
5 | 12x – 8 | - 8 .
4
< 33
13) -5|14 - 7x| + 6 < 41
15)
6 | 7x – 28 | .
14
–3<3
10) 3|8x + 16| - 35 < -5
4
14) -3 |15 - ⅔x| + 35 > 2
16) 4| 5x - 2 | + 3 . > -15
-5
17) 3|2x - 8| + 2 > 4
5
18) 5|10x + 5| -3 < 18
4
19) 2|6x – 24|_ + 5 > 17
-9
20) 6|6x – 9| + 4 + 11 > -15
5
5