Step by step
The question
Question: solve 3|x + 2| − 4 ≤ 5.
1 Get the bars alone
We want the bars by themselves.
The − 4 is in the way, so we add 4 on both sides.
− 4 and + 4 make zero.
Cross them out.
5 plus 4 is 9.
To remove times 3, we divide both sides by 3.
3 is a positive number, so the sign stays the same.
3 divided by 3 is 1.
So just the bars is left.
9 divided by 3 is 3.
2 Draw the distance
3 Write it without bars
So x + 2 is between −3 and 3: −3 ≤ x + 2 ≤ 3.
Now x is in the middle.
4 Remove the + 2
We want x by itself.
The + 2 is in the way.
To remove + 2, we take away 2.
We do it to all three parts.
Why all three?
The signs compare the three parts.
If we change only one part, the answer can change.
+ 2 and − 2 make zero.
Cross them out.
−3 minus 2 is −5.
3 minus 2 is 1.
Now we have −5 ≤ x ≤ 1.
So x is between −5 and 1: −5 ≤ x ≤ 1.
5 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.
6 Check
Let's test one number from the answer: −1.
We put it in the question.
The inside is 1, and the bars make it 1.
−1 is less than or equal to 5.
True!
So the answer works.
Let's look back.
First: get the bars alone.
Then: draw the distance.
Then: write it without bars.
Then: remove the + 2.
Then: show it on a number line.
Last: check.