Step by step
The question
Question: solve 3 − 2x < 1 or x + 5 < 2.
"Or" means one of the parts is enough.
1 Solve the first part
First part: 3 − 2x < 1.
We want x by itself.
Two things are stuck to x: times −2 and + 3.
The + 3 is on the outside, so it comes off first.
Then the times −2.
To remove 3, we take away 3.
We do it on both sides, so the sign stays true.
3 and − 3 make zero.
Cross them out.
1 − 3 = −2.
Now we have −2x < −2.
−2x means −2 times x.
We want just x.
To remove times −2, we divide by −2.
We do it on both sides.
We divide by a negative number.
That turns the sign around.
Less than becomes greater than.
−2 divided by −2 is 1.
So just x is left.
−2 divided by −2 is 1.
So x is greater than 1.
Careful!
Many people forget to turn the sign.
They get x < 1.
But we divided by a negative number.
2 Solve the second part
Second part: x + 5 < 2.
We want x by itself.
The + 5 is in the way.
To remove + 5, we take away 5.
We do it on both sides, so the sign stays true.
+ 5 and − 5 make zero.
Cross them out.
2 − 5 = −3.
So x is less than −3.
3 Show each part on a number line
4 Put the parts together
"Or": we keep every number that is shaded on either line.
So x < −3 or x > 1.
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.
5 Check
Let's test one number from each piece.
First −4.
1 is less than 2.
True!
Now 2, from the other piece.
−1 is less than 1.
True!
So the answer works.
Let's look back.
First: solve the first part.
Then: solve the second part.
Then: show each part on a number line.
Then: put the parts together.
Last: check.