Step by step
The question
Question: solve −x/2 + 4 ≤ 6.
This is an inequality.
The sign means less than or equal to.
1 Remove the fraction
The bottom number is 2.
Times 2 will remove it.
We multiply every part by 2.
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.
2 is a positive number, so the sign stays the same.
2 divided by 2 is 1.
Write what is left.
Do the multiplications.
No more fractions!
2 Remove the + 8
We want x by itself.
The + 8 is in the way.
To remove + 8, we take away 8.
We do it on both sides, so the sign stays true.
+ 8 and − 8 make zero.
Cross them out.
12 − 8 = 4.
Now we have −x ≤ 4.
3 Remove the −1
−x means −1 times x.
We want just x.
To remove times −1, we divide by −1.
We do it on both sides.
We divide by a negative number.
That turns the sign around.
Less than or equal to becomes greater than or equal to.
−1 divided by −1 is 1.
So just x is left.
4 divided by −1 is −4.
So x is greater than or equal to −4.
Careful!
Many people forget to turn the sign.
They get x ≤ −4.
But we divided by a negative number.
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: −3.
We put it in place of x.
First the left side.
Now the right side.
Is 11/2 less than or equal to 6?
To compare fractions, they need the same bottom number: 2.
Multiply the top and bottom of each one by the same number.
Same bottom, so we compare the tops.
11 is less than or equal to 12.
True!
So the answer works.
Let's look back.
First: remove the fraction.
Then: remove the + 8.
Then: remove the −1.
Then: show it on a number line.
Last: check.