What are all the solutions of Step by Step

Step by step

The question

Question: solve |x + 4| = 9.

|x + 4| = 9

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Two cases

The inside, x + 4, is 9 away from 0.

On the number line, 9 and minus 9 are both 9 away from 0.

So the inside is 9, or minus 9.

|A| = 9 → A = 9 or A = −9

9 and −9 are both 9 away from 0

Two small equations, side by side.

x + 4 = 9 or x + 4 = −9

Careful!

Many people solve only x + 4 = 9.

Then they lose the second answer.

Careful: not only x + 4 = 9 ✗

Is x alone now?

Not yet: + 4 is still with x.

How do we remove it?

2 Solve both

In each one, we take away 4 on both sides.

x ~~+ 4 − 4~~ = 9 − 4 or x ~~+ 4 − 4~~ = −9 − 4

+ 4 and − 4 make zero, in both.

9 minus 4 is 5.

−9 minus 4 is −13.

9 − 4 = 5

−9 − 4 = −13

So x = 5 or x = −13.

x = 5 or x = −13

3 Check

Let's check: put x = 5 in.

The inside is 9, and the bars make it 9.

x = 5: |5 + 4| = |9| = 9

Now put x = −13 in.

The inside is −9, and the bars make it 9.

x = −13: |(−13) + 4| = |−9| = 9

Both give 9.

Both answers work!

4 Pick the choice

Our answer is x = 5 or x = −13.

That is choice A.

A) −13 and 5

5 Why not the others

B is −5 and 13.

Wrong: signs swapped: we take 4 away.

B) −5 and 13 ✗ signs swapped: we take 4 away

C is 5 only.

Wrong: forgot the second case: x + 4 can also be −9.

That is the Careful mistake from before.

C) 5 only ✗ forgot the second case: x + 4 can also be −9

D is −9 and 9.

Wrong: forgot the + 4.

D) −9 and 9 ✗ forgot the + 4

Let's look back.

First: two cases.

Then: solve both.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer