Step by step
The question
Question: solve −2x + 1 ≥ 7 or x − 4 > 0.
"Or" means one of the parts is enough.
1 Solve the first part
First part: −2x + 1 ≥ 7.
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.
7 − 1 = 6.
Now we have −2x ≥ 6.
−2x means −2 times x.
We want just x.
To remove times −2, we divide by −2.
We do it on both sides.
We divide by a negative number.
That turns the sign around.
Greater than or equal to becomes less than or equal to.
−2 divided by −2 is 1.
So just x is left.
6 divided by −2 is −3.
So x is less than or equal to −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: x − 4 > 0.
We want x by itself.
The − 4 is in the way.
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.
0 + 4 = 4.
So x is greater than 4.
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 > 4.
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 −4.
9 is greater than or equal to 7.
True!
Now 5, from the other piece.
1 is greater than 0.
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.