Step by step
The question
Question: solve −x + 2 > 4 or 3x − 1 ≥ 8.
"Or" means one of the parts is enough.
1 Solve the first part
First part: −x + 2 > 4.
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.
4 − 2 = 2.
Now we have −x > 2.
−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.
2 divided by −1 is −2.
So x is less than −2.
Careful!
Many people forget to turn the sign.
They get x > −2.
But we divided by a negative number.
2 Solve the second part
Second part: 3x − 1 ≥ 8.
We want x by itself.
Two things are stuck to x: times 3 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 3.
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.
8 + 1 = 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 greater than or equal to 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 < −2 or x ≥ 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.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
∪ means or.
5 Check
Let's test one number from each piece.
First −3.
5 is greater than 4.
True!
Now 4, from the other piece.
11 is greater than or equal to 8.
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.