Step by step
The question
Question: solve |2x − 3| = 7.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Two cases
The inside, 2x − 3, 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.
Two small equations, side by side.
Careful!
Many people solve only 2x − 3 = 7.
Then they lose the second answer.
Is x alone now?
Not yet: − 3 is still with x.
How do we remove it?
2 Solve both
In each one, we add 3 on both sides.
− 3 and + 3 make zero, in both.
7 plus 3 is 10.
−7 plus 3 is −4.
Now we have 2x = 10 or 2x = −4.
To remove times 2, we divide by 2.
10 divided by 2 is 5.
−4 divided by 2 is −2.
So x = 5 or x = −2.
3 Check
Let's check: put x = 5 in.
The inside is 7, and the bars make it 7.
Now put x = −2 in.
The inside is −7, and the bars make it 7.
Both give 7.
Both answers work!
4 Pick the choice
Our answer is x = 5 or x = −2.
That is choice B.
5 Why not the others
A is 5 only.
Wrong: forgot the second case: 2x − 3 can also be −7.
That is the Careful mistake from before.
C is −5 and 2.
Wrong: used + 3 instead of − 3.
D is 2 and 5.
Wrong: 2x = −4 gives x = −2, not 2.
Let's look back.
First: two cases.
Then: solve both.
Then: check.
Then: pick the choice.
Last: why not the others.