Step by step
The question
Question: solve 5 − x > 2 and 2x + 3 > −1.
"And" means both parts must be true.
1 Solve the first part
First part: 5 − x > 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.
Now we have −x > −3.
−x means −1 times x.
We want just x.
To remove times −1, we divide by −1.
We do it on both sides.
We divide by a negative number.
That turns the sign around.
Greater than becomes less than.
−1 divided by −1 is 1.
So just x is left.
−3 divided by −1 is 3.
So x is less than 3.
Careful!
Many people forget to turn the sign.
They get x > 3.
But we divided by a negative number.
2 Solve the second part
Second part: 2x + 3 > −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 = −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.
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.
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.
4 is greater than 2.
True!
Now the second part.
5 is greater 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: find the numbers in both.
Last: check.