Step by step
The question
Question: solve 3(2 − x) < x + 2.
This is an inequality.
The sign means less than.
1 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
Work out each multiplication.
2 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the −3x from the left side.
To remove −3x, we add 3x.
We do it on both sides.
Why both sides?
The sign compares the two sides.
If we change only one side, the answer can change.
− 3x and + 3x make zero.
Cross them out.
x + 3x = 4x.
Now we have 6 < 4x + 2.
Let's swap the two sides, so x is on the left.
When we swap, the sign turns around too.
Less than becomes greater than.
We want x by itself.
Two things are stuck to x: times 4 and + 2.
The + 2 is on the outside, so it comes off first.
Then the times 4.
3 Remove the + 2
To remove + 2, we take away 2.
We do it on both sides, so the sign stays true.
+ 2 and − 2 make zero.
Cross them out.
6 − 2 = 4.
Now we have 4x > 4.
4 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so the sign stays true.
4 divided by 4 is 1.
So just x is left.
4 divided by 4 is 1.
So x is greater than 1.
5 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A round bracket means the end is not in, like the open circle.
∞ means it goes on for ever.
6 Check
Let's test one number from the answer: 2.
We put it in place of x.
First the left side.
Now the right side.
0 is less than 4.
True!
So the answer works.
Let's look back.
First: open the brackets.
Then: get x on one side.
Then: remove the + 2.
Then: remove the 4.
Then: show it on a number line.
Last: check.