Step by step
The question
Question: solve x − 1 < 2 or x − 1 > 5.
Two inequalities, joined by "or".
x − 1 < 2 or x − 1 > 5
"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 − 1
We want x by itself.
The − 1 is in the way.
In each one, we add 1 on both sides.
x − 1 + 1 < 2 + 1 or x − 1 + 1 > 5 + 1
− 1 and + 1 make zero, in both.
2 plus 1 is 3.
5 plus 1 is 6.
2 + 1 = 3, 5 + 1 = 6
So x < 3 or x > 6.
x < 3 or x > 6
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: (−∞, 3) ∪ (6, ∞)
3 Check
Let's test one number from each part.
First 2, from the left part.
x = 2: 2 − 1 = 1
1 is less than 2.
True!
1 < 2
Now 7, from the right part.
x = 7: 7 − 1 = 6
6 is greater than 5.
True!
Both parts work!
6 > 5
Let's look back.
First: remove the − 1.
Then: show it on a number line.
Last: check.
Answer
x < 3 or x > 6