Step by step
The question
Question: solve 3x + 2 ≤ x + 8.
This is an inequality.
The sign means less than or equal to.
1 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the x from the right side.
To remove x, we take away x.
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.
x and − x make zero.
Cross them out.
3x − x = 2x.
Now we have 2x + 2 ≤ 8.
We want x by itself.
Two things are stuck to x: times 2 and + 2.
The + 2 is on the outside, so it comes off first.
Then the times 2.
2 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.
8 − 2 = 6.
Now we have 2x ≤ 6.
3 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so the sign stays true.
2 divided by 2 is 1.
So just x is left.
6 divided by 2 is 3.
So x is less than or equal to 3.
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: 2.
We put it in place of x.
First the left side.
Now the right side.
8 is less than or equal to 10.
True!
So the answer works.
Let's look back.
First: get x on one side.
Then: remove the + 2.
Then: remove the 2.
Then: show it on a number line.
Last: check.