Step by step
The question
Question: solve |x − 3| = 5.
|x − 3| = 5
1 Two cases
The inside is 5 away from 0.
On the number line, 5 and minus 5 are both 5 away from 0.
So the inside is 5, or minus 5.
|A| = 5 → A = 5 or A = −5
5 and −5 are both 5 away from 0
Two small equations, side by side.
x − 3 = 5 or x − 3 = −5
Careful!
Many people solve only x − 3 = 5.
Then they lose the second answer.
Careful: not only x − 3 = 5 ✗
2 Solve both
In each one, we add 3 on both sides.
x − 3 + 3 = 5 + 3 or x − 3 + 3 = −5 + 3
− 3 and + 3 make zero, in both.
So x = 8 or x = −2.
x = 8 or x = −2
3 Check
Let's check: put x = 8 in.
The inside is 5, and the bars make it 5.
x = 8: |8 − 3| = |5| = 5
Now put x = −2 in.
The inside is −5, and the bars make it 5.
x = −2: |(−2) − 3| = |−5| = 5
Both give 5.
Both answers work!
Let's look back.
First: two cases.
Then: solve both.
Last: check.
Answer
x = 8 or x = −2