Solve -2|x + 3| + 10 = 4 Step by Step | Absolute value equations

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The question

Question: solve −2|x + 3| + 10 = 4.

−2|x + 3| + 10 = 4

1 Get the bars alone

We want the bars by themselves.

The + 10 is in the way, so we take away 10 on both sides.

−2|x + 3| ~~+ 10 − 10~~ = 4 − 10

+ 10 and − 10 make zero.

Cross them out.

4 minus 10 is −6.

−2|x + 3| = −6

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

−2|x + 3|−2 = −6−2

−2 divided by −2 is 1.

So just the bars is left.

−6 divided by −2 is 3.

|x + 3| = 3

2 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.

x + 3 = 3 or x + 3 = −3

Careful!

Many people solve only x + 3 = 3.

Then they lose the second answer.

Careful: not only x + 3 = 3 ✗

3 Solve both

In each one, we take away 3 on both sides.

x ~~+ 3 − 3~~ = 3 − 3 or x ~~+ 3 − 3~~ = −3 − 3

+ 3 and − 3 make zero, in both.

So x = 0 or x = −6.

x = 0 or x = −6

4 Check

Let's check: put x = 0 in.

The inside is 3, and the bars make it 3.

x = 0: −2|0 + 3| + 10 = −2|3| + 10

= −2(3) + 10 = 4

Now put x = −6 in.

The inside is −3, and the bars make it 3.

x = −6: −2|(−6) + 3| + 10 = −2|−3| + 10

= −2(3) + 10 = 4

Both give 4.

Both answers work!

Let's look back.

First: get the bars alone.

Then: two cases.

Then: solve both.

Last: check.

Answer