Step by step
The question
Question: solve −2(x + 3) ≤ 4 − x.
This is an inequality.
The sign means less than or equal to.
1 Open the brackets
The −2 in front of the bracket multiplies every part inside.
So we write −2 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 −2x from the left side.
To remove −2x, we add 2x.
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.
− 2x and + 2x make zero.
Cross them out.
−x + 2x = x.
Now we have −6 ≤ 4 + x.
Let's swap the two sides, so x is on the left.
When we swap, the sign turns around too.
Less than or equal to becomes greater than or equal to.
3 Remove the 4
We want x by itself.
The 4 is in the way.
To remove 4, we take away 4.
We do it on both sides, so the sign stays true.
4 and − 4 make zero.
Cross them out.
−6 − 4 = −10.
So x is greater than or equal to −10.
4 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
5 Check
Let's test one number from the answer: −9.
We put it in place of x.
First the left side.
Now the right side.
12 is less than or equal to 13.
True!
So the answer works.
Let's look back.
First: open the brackets.
Then: get x on one side.
Then: remove the 4.
Then: show it on a number line.
Last: check.