Step by step
The question
Question: solve 2x + 1 ≤ 9.
This is an inequality.
The sign means less than or equal to.
Let's find x.
We want x by itself.
First we remove the + 1, so 2x is alone.
Then we remove the times 2.
Is x alone now?
Not yet: + 1 is still with x.
How do we remove it?
1 Remove the + 1
To remove + 1, we take away 1.
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.
+ 1 and − 1 cancel each other.
They make zero.
9 − 1 = 8.
Now we have 2x ≤ 8.
So after step 1: we took away 1, and now we have 2x ≤ 8.
Is x alone now?
Not yet: 2x means 2 times x.
2 Remove the 2
To remove times 2, we divide by 2.
We do it on both sides, so the sign stays the same.
2 divided by 2 is 1.
They cancel each other.
So just x is left.
8 divided by 2 is 4.
So x is less than or equal to 4.
3 Show it on a number line
A filled dot at 4.
4 is part of the answer, because of the "or equal to".
We shade to the left.
These are all the numbers less than or equal to 4.
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.
−∞ goes on for ever to the left, ∞ to the right.
4 Check
Let's test one number from the answer: 3.
3 is in the answer: it is less than or equal to 4.
We put it in place of x.
First the left side.
Now the right side.
7 is less than or equal to 9.
True!
So the answer works.
Let's look back.
First: remove the + 1.
Then: remove the 2.
Then: show it on a number line.
Last: check.