Step by step
The question
Question: solve x − 4 < −2 or x − 4 > 3.
Two inequalities, joined by "or".
x − 4 < −2 or x − 4 > 3
"Or" means: x can make one of them true, or the other.
We solve both, side by side.
or: one of them is true
1 Remove the − 4
We want x by itself.
The − 4 is in the way.
In each one, we add 4 on both sides.
x − 4 + 4 < −2 + 4 or x − 4 + 4 > 3 + 4
− 4 and + 4 make zero, in both.
−2 plus 4 is 2.
3 plus 4 is 7.
−2 + 4 = 2, 3 + 4 = 7
So x < 2 or x > 7.
x < 2 or x > 7
2 Show it on a number line
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.
∪ means or.
interval: (−∞, 2) ∪ (7, ∞)
3 Check
Let's test one number from each part.
First 1, from the left part.
x = 1: 1 − 4 = −3
−3 is less than −2.
True!
−3 < −2
Now 8, from the right part.
x = 8: 8 − 4 = 4
4 is greater than 3.
True!
Both parts work!
4 > 3
Let's look back.
First: remove the − 4.
Then: show it on a number line.
Last: check.
Answer
x < 2 or x > 7