Step by step
The question
Question: solve |x + 4| − 2 = 5.
|x + 4| − 2 = 5
1 Get the bars alone
We want the bars by themselves.
The − 2 is in the way, so we add 2 on both sides.
|x + 4| − 2 + 2 = 5 + 2
− 2 and + 2 make zero.
Cross them out.
5 plus 2 is 7.
|x + 4| = 7
2 Two cases
The inside is 7 away from 0.
On the number line, 7 and minus 7 are both 7 away from 0.
So the inside is 7, or minus 7.
|A| = 7 → A = 7 or A = −7
7 and −7 are both 7 away from 0
Two small equations, side by side.
x + 4 = 7 or x + 4 = −7
Careful!
Many people solve only x + 4 = 7.
Then they lose the second answer.
Careful: not only x + 4 = 7 ✗
3 Solve both
In each one, we take away 4 on both sides.
x + 4 − 4 = 7 − 4 or x + 4 − 4 = −7 − 4
+ 4 and − 4 make zero, in both.
So x = 3 or x = −11.
x = 3 or x = −11
4 Check
Let's check: put x = 3 in.
The inside is 7, and the bars make it 7.
x = 3: |3 + 4| − 2 = |7| − 2 = 7 − 2 = 5
Now put x = −11 in.
The inside is −7, and the bars make it 7.
x = −11: |(−11) + 4| − 2 = |−7| − 2 = 7 − 2
= 5
Both give 5.
Both answers work!
Let's look back.
First: get the bars alone.
Then: two cases.
Then: solve both.
Last: check.
Answer
x = 3 or x = −11