Step by step
The question
Question: solve 5x − 3 > 2x + 9.
This is an inequality.
The sign means greater than.
Let's find x.
1 Get x on one side
There is x on both sides.
We want x on one side only.
The numbers in front of x are 2 and 5.
We remove the smaller one, 2x, so the x part that is left is plus.
So we remove the 2x from the right side.
To remove 2x, we take away 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 cancel each other.
They make zero.
5x − 2x = 3x.
Now we have 3x − 3 > 9.
We want x by itself.
First we remove the − 3, so 3x is alone.
Then we remove the times 3.
So after step 1: we took away 2x, and now we have 3x − 3 > 9.
Is x alone now?
Not yet: − 3 is still with x.
How do we remove it?
2 Remove the − 3
To remove − 3, we add 3.
We do it on both sides, so the sign stays the same.
− 3 and + 3 cancel each other.
They make zero.
9 + 3 = 12.
Now we have 3x > 12.
So after step 2: we added 3, and now we have 3x > 12.
Is x alone now?
Not yet: 3x means 3 times x.
3 Remove the 3
To remove times 3, we divide by 3.
We do it on both sides, so the sign stays the same.
3 divided by 3 is 1.
They cancel each other.
So just x is left.
12 divided by 3 is 4.
So x is greater than 4.
4 Show it on a number line
An open circle at 4.
4 itself is not part of the answer.
We shade to the right.
These are all the numbers greater than 4.
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.
−∞ goes on for ever to the left, ∞ to the right.
5 Check
Let's test one number from the answer: 5.
5 is in the answer: it is greater than 4.
We put it in place of x.
First the left side.
Now the right side.
22 is greater than 19.
True!
So the answer works.
Let's look back.
First: get x on one side.
Then: remove the − 3.
Then: remove the 3.
Then: show it on a number line.
Last: check.