Solve |n - 4| = |2n + 1|

Step by step

The question

Question: solve |n − 4| = |2n + 1|.

Both bars give how far from 0.

Two numbers the same distance from 0 are equal, or opposite.

Like 3 and minus 3.

Careful!

The minus is for the whole of 2n + 1.

Every sign in the bracket changes.

1 Two cases

So n − 4 = 2n + 1, or n − 4 = −(2n + 1).

We keep the bracket: the minus is for all of it.

2 Solve the first case

First case: n − 4 = 2n + 1.

3 Get n on one side

There is n on both sides.

We want n on one side only.

So we remove the n from the left side.

To remove n, we take away n.

We do it on both sides, so they stay equal.

n and − n make zero.

Cross them out.

2n − n = n.

Now we have −4 = n + 1.

Let's swap the two sides, so n is on the left.

4 Remove the + 1

We want n by itself.

The + 1 is in the way.

To remove + 1, we take away 1.

We do it on both sides, so they stay equal.

+ 1 and − 1 make zero.

Cross them out.

−4 − 1 = −5.

So n = −5!

5 Solve the second case

Second case: n − 4 = −(2n + 1).

6 Open the brackets

A minus in front of a bracket changes the sign of every part inside.

2n becomes −2n.

1 becomes −1.

7 Get n on one side

There is n on both sides.

We want n on one side only.

So we remove the −2n from the right side.

To remove −2n, we add 2n.

We do it on both sides, so they stay equal.

− 2n and + 2n make zero.

Cross them out.

n + 2n = 3n.

Now we have 3n − 4 = −1.

We want n by itself.

Two things are stuck to n: times 3 and − 4.

The − 4 is on the outside, so it comes off first.

Then the times 3.

8 Remove the − 4

To remove − 4, we add 4.

We do it on both sides, so they stay equal.

− 4 and + 4 make zero.

Cross them out.

−1 + 4 = 3.

Now we have 3n = 3.

9 Remove the 3

3n means 3 times n.

We want just n.

To remove times 3, we divide by 3.

We do it on both sides, so they stay equal.

3 divided by 3 is 1.

So just n is left.

3 divided by 3 is 1.

So n = 1!

10 Write the answers

So n = −5 or n = 1.

11 Check

Let's check: put n = −5 in.

The left bars give 9.

The right bars give 9 too.

The same!

Now put n = 1 in.

The left bars give 3.

The right bars give 3 too.

The same!

Both answers work!

Let's look back.

First: two cases.

Then: solve the first case.

Then: get n on one side.

Then: remove the + 1.

Then: solve the second case.

Then: open the brackets.

Then: get n on one side.

Then: remove the − 4.

Then: remove the 3.

Then: write the answers.

Last: check.