Step by step
The question
Question: solve |x/2 − 1| = 3.
|x2 − 1| = 3
1 Two cases
The inside is 3 away from 0.
On the number line, 3 and minus 3 are both 3 away from 0.
So the inside is 3, or minus 3.
|A| = 3 → A = 3 or A = −3
3 and −3 are both 3 away from 0
Two small equations, side by side.
x2 − 1 = 3 or x2 − 1 = −3
Careful!
Many people solve only x/2 − 1 = 3.
Then they lose the second answer.
Careful: not only x2 − 1 = 3 ✗
2 Solve both
In each one, we add 1 on both sides.
x2 − 1 + 1 = 3 + 1 or x2 − 1 + 1 = −3 + 1
− 1 and + 1 make zero, in both.
3 plus 1 is 4.
−3 plus 1 is −2.
x2 = 4 or x2 = −2
The bottom number is 2.
Times 2 will remove it.
In each one, we multiply both sides by 2.
2/2 is 1.
The fraction is gone, in both.
2 times 4 is 8.
2 times −2 is −4.
So x = 8 or x = −4.
x = 8 or x = −4
3 Check
Let's check: put x = 8 in.
The inside is 3, and the bars make it 3.
x = 8: |82 − 1| = |3| = 3
Now put x = −4 in.
The inside is −3, and the bars make it 3.
x = −4: |(−4)2 − 1| = |−3| = 3
Both give 3.
Both answers work!
Let's look back.
First: two cases.
Then: solve both.
Last: check.
Answer
x = 8 or x = −4