Step by step
The question
Question: solve 2x + 1 > −3 and 3x − 2 ≤ 7.
"And" means both parts must be true.
1 Solve the first part
First part: 2x + 1 > −3.
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.
−3 − 1 = −4.
Now we have 2x > −4.
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.
−4 divided by 2 is −2.
So x is greater than −2.
2 Solve the second part
Second part: 3x − 2 ≤ 7.
We want x by itself.
Two things are stuck to x: times 3 and − 2.
The − 2 is on the outside, so it comes off first.
Then the times 3.
To remove − 2, we add 2.
We do it on both sides, so the sign stays true.
− 2 and + 2 make zero.
Cross them out.
7 + 2 = 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 or equal to 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 −2 < x ≤ 3.
x is between −2 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.
A square bracket means the end is in, like the filled circle.
5 Check
Let's test one number from the answer: 1.
"And": it must work in both parts.
First part.
3 is greater than −3.
True!
Now the second part.
1 is less than or equal to 7.
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.