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Solve |2y + 3| = |y - 6|

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Step by step

The question

Question: solve |2y + 3| = |y − 6|.

|2y + 3| = |y − 6|

Both bars give how far from 0.

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

Like 3 and minus 3.

|A| = |B| → A = B or A = −B
|3| = |−3|: 3 and −3 are both 3 away from 0

Careful!

The minus is for the whole of y − 6.

Every sign in the bracket changes.

Careful: not 2y + 3 = −y − 6 ✗

1 Two cases

So 2y + 3 = y − 6, or 2y + 3 = −(y − 6).

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

2y + 3 = y − 6 or 2y + 3 = −(y − 6)

2 Solve the first case

First case: 2y + 3 = y − 6.

2y + 3 = y − 6

3 Get y on one side

There is y on both sides.

We want y on one side only.

So we remove the y from the right side.

To remove y, we take away y.

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

2y + 3 − y = y − 6 − y

y and − y make zero.

Cross them out.

2y − y = y.

2y − y = y

Now we have y + 3 = −6.

y + 3 = −6

4 Remove the + 3

We want y by itself.

The + 3 is in the way.

To remove + 3, we take away 3.

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

y + 3 − 3 = −6 − 3

+ 3 and − 3 make zero.

Cross them out.

−6 − 3 = −9.

−6 − 3 = −9

So y = −9!

y = −9

5 Solve the second case

Second case: 2y + 3 = −(y − 6).

2y + 3 = −(y − 6)

6 Open the brackets

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

y becomes −y.

−6 becomes +6.

2y + 3 = −y + 6

7 Get y on one side

There is y on both sides.

We want y on one side only.

So we remove the −y from the right side.

To remove −y, we add y.

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

2y + 3 + y = −y + 6 + y

− y and + y make zero.

Cross them out.

2y + y = 3y.

2y + y = 3y

Now we have 3y + 3 = 6.

3y + 3 = 6

We want y by itself.

Two things are stuck to y: times 3 and + 3.

The + 3 is on the outside, so it comes off first.

Then the times 3.

8 Remove the + 3

To remove + 3, we take away 3.

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

3y + 3 − 3 = 6 − 3

+ 3 and − 3 make zero.

Cross them out.

6 − 3 = 3.

6 − 3 = 3

Now we have 3y = 3.

3y = 3

9 Remove the 3

3y means 3 times y.

We want just y.

To remove times 3, we divide by 3.

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

3y3 = 33

3 divided by 3 is 1.

So just y is left.

3 divided by 3 is 1.

33 = 1

So y = 1!

y = 1

10 Write the answers

So y = −9 or y = 1.

y = −9 or y = 1

11 Check

Let's check: put y = −9 in.

The left bars give 15.

y = −9: |2(−9) + 3| = |−18 + 3| = |−15|
= 15

The right bars give 15 too.

The same!

|−9 − 6| = |−15| = 15

Now put y = 1 in.

The left bars give 5.

y = 1: |2(1) + 3| = |2 + 3| = |5| = 5

The right bars give 5 too.

The same!

|1 − 6| = |−5| = 5

Both answers work!

Let's look back.

First: two cases.

Then: solve the first case.

Then: get y on one side.

Then: remove the + 3.

Then: solve the second case.

Then: open the brackets.

Then: get y on one side.

Then: remove the + 3.

Then: remove the 3.

Then: write the answers.

Last: check.

Answer
y = −9 or y = 1

Your turn

Try this one yourself, then check:

Solve |2y + 4| = |y − 1|
Show the answer
y = −5 or y = −1

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