Step by step
The question
Question: solve 4x − 3 > 1 and x + 6 < 10.
"And" means both parts must be true.
1 Solve the first part
First part: 4x − 3 > 1.
We want x by itself.
Two things are stuck to x: times 4 and − 3.
The − 3 is on the outside, so it comes off first.
Then the times 4.
To remove − 3, we add 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 4x > 4.
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so the sign stays true.
4 divided by 4 is 1.
So just x is left.
4 divided by 4 is 1.
So x is greater than 1.
2 Solve the second part
Second part: x + 6 < 10.
We want x by itself.
The + 6 is in the way.
To remove + 6, we take away 6.
We do it on both sides, so the sign stays true.
+ 6 and − 6 make zero.
Cross them out.
10 − 6 = 4.
So x is less than 4.
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 1 < x < 4.
x is between 1 and 4.
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.
5 Check
Let's test one number from the answer: 3.
"And": it must work in both parts.
First part.
9 is greater than 1.
True!
Now the second part.
9 is less than 10.
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.