Step by step
The question
Question: solve x/2 + x/3 > 5.
This is an inequality.
The sign means greater than.
1 Remove the fractions
The bottom numbers are 2 and 3.
6 can be divided by all of them.
We multiply every part by 6.
We do it on both sides.
Why both sides?
The sign compares the two sides.
If we change only one side, the answer can change.
6 is a positive number, so the sign stays the same.
6 divided by 2 is 3.
6 divided by 3 is 2.
Write what is left.
Do the multiplications.
No more fractions!
2 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
3x + 2x = 5x.
Now we have 5x > 30.
3 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so the sign stays true.
5 divided by 5 is 1.
So just x is left.
30 divided by 5 is 6.
So x is greater than 6.
4 Show it on a number line
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.
∞ means it goes on for ever.
5 Check
Let's test one number from the answer: 7.
We put it in place of x.
First the left side.
Now the right side.
Is 35/6 greater than 5?
To compare fractions, they need the same bottom number: 6.
Multiply the top and bottom of each one by the same number.
Same bottom, so we compare the tops.
35 is greater than 30.
True!
So the answer works.
Let's look back.
First: remove the fractions.
Then: group like terms.
Then: remove the 5.
Then: show it on a number line.
Last: check.