Step by step
The question
Question: solve 0.5x − 1.5 ≥ 2.
This is an inequality.
The sign means greater than or equal to.
1 Remove the decimals
Decimals are hard to work with.
Let's get rid of them.
We multiply every part by 10.
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.
10 is a positive number, so the sign stays the same.
Times 10 moves the point 1 place to the right.
10 × 0.5x = 5x.
10 × 1.5 = 15.
10 × 2 = 20.
Write the answers.
No more decimals!
We want x by itself.
Two things are stuck to x: times 5 and − 15.
The − 15 is on the outside, so it comes off first.
Then the times 5.
2 Remove the − 15
To remove − 15, we add 15.
We do it on both sides, so the sign stays true.
− 15 and + 15 make zero.
Cross them out.
20 + 15 = 35.
Now we have 5x ≥ 35.
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.
35 divided by 5 is 7.
So x is greater than or equal to 7.
4 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
5 Check
Let's test one number from the answer: 8.
We put it in place of x.
First the left side.
Now the right side.
2.5 is greater than or equal to 2.
True!
So the answer works.
Let's look back.
First: remove the decimals.
Then: remove the − 15.
Then: remove the 5.
Then: show it on a number line.
Last: check.