Step by step
The question
Question: solve 7 − x ≥ 3x − 1.
This is an inequality.
The sign means greater 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 left side.
To remove −x, we add 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 = 4x.
Now we have 7 ≥ 4x − 1.
Let's swap the two sides, so x is on the left.
When we swap, the sign turns around too.
Greater than or equal to becomes less than or equal to.
We want x by itself.
Two things are stuck to x: times 4 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 4.
2 Remove the − 1
To remove − 1, we add 1.
We do it on both sides, so the sign stays true.
− 1 and + 1 make zero.
Cross them out.
7 + 1 = 8.
Now we have 4x ≤ 8.
3 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.
8 divided by 4 is 2.
So x is less than or equal to 2.
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: 1.
We put it in place of x.
First the left side.
Now the right side.
6 is greater than or equal to 2.
True!
So the answer works.
Let's look back.
First: get x on one side.
Then: remove the − 1.
Then: remove the 4.
Then: show it on a number line.
Last: check.