Step by step
The question
Question: solve 2(x − 1) ≤ 4 and x + 3 > 1.
"And" means both parts must be true.
1 Solve the first part
First part: 2(x − 1) ≤ 4.
The 2 multiplies the whole bracket.
Easy way: remove the 2 first.
To remove times 2, we divide by 2.
We do it on both sides, so the sign stays true.
2 is a positive number, so the sign stays the same.
2 divided by 2 is 1.
So just the bracket is left.
4 divided by 2 is 2.
Now we have x − 1 ≤ 2.
We don't need the bracket any more.
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.
2 + 1 = 3.
So x is less than or equal to 3.
2 Solve the second part
Second part: x + 3 > 1.
We want x by itself.
The + 3 is in the way.
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 = −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.
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.
0 is less than or equal to 4.
True!
Now the second part.
4 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.