Step by step
The question
Question: solve 3x + 2 < 2x + 5 and x − 1 > −4.
"And" means both parts must be true.
1 Solve the first part
First part: 3x + 2 < 2x + 5.
There is x on both sides.
We want x on one side only.
So we remove the 2x from the right side.
To remove 2x, we take away 2x.
We do it on both sides, so the sign stays true.
2x and − 2x make zero.
Cross them out.
3x − 2x = x.
Now we have x + 2 < 5.
We want x by itself.
The + 2 is in the way.
To remove + 2, we take away 2.
We do it on both sides, so the sign stays true.
+ 2 and − 2 make zero.
Cross them out.
5 − 2 = 3.
So x is less than 3.
2 Solve the second part
Second part: x − 1 > −4.
We want x by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it on both sides, so the sign stays true.
− 1 and + 1 make zero.
Cross them out.
−4 + 1 = −3.
So x is greater than −3.
3 Show each part on a number line
4 Find the numbers in both
"And": we keep only the numbers that are shaded on both lines.
So −3 < x < 3.
x is between −3 and 3.
Careful!
A number that is shaded on only one line is not an answer.
It must be on both.
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.
5 Check
Let's test one number from the answer: 1.
"And": it must work in both parts.
First part.
5 is less than 7.
True!
Now the second part.
0 is greater than −4.
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: find the numbers in both.
Last: check.