Step by step
The question
Question: solve 3x − 4 < 11.
This is an inequality.
The sign means less than.
Let's find x.
We want x by itself.
First we remove the − 4, so 3x is alone.
Then we remove the times 3.
Is x alone now?
Not yet: − 4 is still with x.
How do we remove it?
1 Remove the − 4
To remove − 4, we add 4.
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.
− 4 and + 4 cancel each other.
They make zero.
11 + 4 = 15.
Now we have 3x < 15.
So after step 1: we added 4, and now we have 3x < 15.
Is x alone now?
Not yet: 3x means 3 times x.
2 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.
15 divided by 3 is 5.
So x is less than 5.
3 Show it on a number line
An open circle at 5.
5 itself is not part of the answer.
We shade to the left.
These are all the numbers less than 5.
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.
4 Check
Let's test one number from the answer: 4.
4 is in the answer: it is less than 5.
We put it in place of x.
First the left side.
Now the right side.
8 is less than 11.
True!
So the answer works.
Let's look back.
First: remove the − 4.
Then: remove the 3.
Then: show it on a number line.
Last: check.