Step by step
The question
Question: solve 3x − 4 < 5 or 2x + 1 > 11.
"Or" means one of the parts is enough.
1 Solve the first part
First part: 3x − 4 < 5.
We want x by itself.
Two things are stuck to x: times 3 and − 4.
The − 4 is on the outside, so it comes off first.
Then the times 3.
To remove − 4, we add 4.
We do it on both sides, so the sign stays true.
− 4 and + 4 make zero.
Cross them out.
5 + 4 = 9.
Now we have 3x < 9.
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so the sign stays true.
3 divided by 3 is 1.
So just x is left.
9 divided by 3 is 3.
So x is less than 3.
2 Solve the second part
Second part: 2x + 1 > 11.
We want x by itself.
Two things are stuck to x: times 2 and + 1.
The + 1 is on the outside, so it comes off first.
Then the times 2.
To remove + 1, we take away 1.
We do it on both sides, so the sign stays true.
+ 1 and − 1 make zero.
Cross them out.
11 − 1 = 10.
Now we have 2x > 10.
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so the sign stays true.
2 divided by 2 is 1.
So just x is left.
10 divided by 2 is 5.
So x is greater than 5.
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 > 5.
Careful!
Many people write 5 < x < 3.
But a chain means between, and no number is between like that.
So we keep the word "or".
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 2.
2 is less than 5.
True!
Now 6, from the other piece.
13 is greater than 11.
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.