Solve 2|x - 1| + 3 = 11 Step by Step | Absolute value equations

Step by step

The question

Question: solve 2|x − 1| + 3 = 11.

2|x − 1| + 3 = 11

1 Get the bars alone

We want the bars by themselves.

The + 3 is in the way, so we take away 3 on both sides.

2|x − 1| ~~+ 3 − 3~~ = 11 − 3

+ 3 and − 3 make zero.

Cross them out.

11 minus 3 is 8.

2|x − 1| = 8

To remove times 2, we divide both sides by 2.

2|x − 1|2 = 82

2 divided by 2 is 1.

So just the bars is left.

8 divided by 2 is 4.

|x − 1| = 4

2 Two cases

The inside is 4 away from 0.

On the number line, 4 and minus 4 are both 4 away from 0.

So the inside is 4, or minus 4.

|A| = 4 → A = 4 or A = −4

4 and −4 are both 4 away from 0

Two small equations, side by side.

x − 1 = 4 or x − 1 = −4

Careful!

Many people solve only x − 1 = 4.

Then they lose the second answer.

Careful: not only x − 1 = 4 ✗

3 Solve both

In each one, we add 1 on both sides.

x ~~− 1 + 1~~ = 4 + 1 or x ~~− 1 + 1~~ = −4 + 1

− 1 and + 1 make zero, in both.

So x = 5 or x = −3.

x = 5 or x = −3

4 Check

Let's check: put x = 5 in.

The inside is 4, and the bars make it 4.

x = 5: 2|5 − 1| + 3 = 2|4| + 3 = 2(4) + 3

= 11

Now put x = −3 in.

The inside is −4, and the bars make it 4.

x = −3: 2|(−3) − 1| + 3 = 2|−4| + 3

= 2(4) + 3 = 11

Both give 11.

Both answers work!

Let's look back.

First: get the bars alone.

Then: two cases.

Then: solve both.

Last: check.

Answer