Worked examples
The question
Question: solve |x| < 4.
|x| < 4
1 Draw the distance
2 Write it without bars
So x is between −4 and 4: −4 < x < 4.
Now x is in the middle.
−4 < x < 4
So x is between −4 and 4: −4 < x < 4.
3 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.
interval: (−4, 4)
4 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.
x = 1: |1| = 1
1 is less than 4.
True!
So the answer works.
1 < 4
Let's look back.
First: draw the distance.
Then: write it without bars.
Then: show it on a number line.
Last: check.
▶ Watch it solved
The question
Question: solve |2x − 1| ≤ 5.
|2x − 1| ≤ 5
1 Draw the distance
2 Write it without bars
So 2x − 1 is between −5 and 5: −5 ≤ 2x − 1 ≤ 5.
Now x is in the middle.
−5 ≤ 2x − 1 ≤ 5
3 Remove the − 1
We want x by itself.
The − 1 is in the way.
To remove − 1, we add 1.
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.
−5 + 1 ≤ 2x − 1 + 1 ≤ 5 + 1
− 1 and + 1 make zero.
Cross them out.
−5 plus 1 is −4.
5 plus 1 is 6.
−5 + 1 = −4
5 + 1 = 6
Now we have −4 ≤ 2x ≤ 6.
−4 ≤ 2x ≤ 6
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it to all three parts.
2 is a positive number, so the signs stay the same.
−42 ≤ 2x2 ≤ 62
2 divided by 2 is 1.
So just x is left.
−4 divided by 2 is −2.
6 divided by 2 is 3.
−42 = −2, 62 = 3
Now we have −2 ≤ x ≤ 3.
So x is between −2 and 3: −2 ≤ x ≤ 3.
−2 ≤ x ≤ 3
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.
interval: [−2, 3]
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.
x = 1: |2(1) − 1| = |2 − 1| = |1| = 1
1 is less than or equal to 5.
True!
So the answer works.
1 ≤ 5
Let's look back.
First: draw the distance.
Then: write it without bars.
Then: remove the − 1.
Then: remove the 2.
Then: show it on a number line.
Last: check.
▶ Watch it solved
The question
Question: solve 3|x + 2| − 4 ≤ 5.
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.
3|x + 2| − 4 + 4 ≤ 5 + 4
− 4 and + 4 make zero.
Cross them out.
5 plus 4 is 9.
3|x + 2| ≤ 9
To remove times 3, we divide both sides by 3.
3 is a positive number, so the sign stays the same.
3|x + 2|3 ≤ 93
3 divided by 3 is 1.
So just the bars is left.
9 divided by 3 is 3.
|x + 2| ≤ 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.
−3 ≤ x + 2 ≤ 3
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.
−3 − 2 ≤ x + 2 − 2 ≤ 3 − 2
+ 2 and − 2 make zero.
Cross them out.
−3 minus 2 is −5.
3 minus 2 is 1.
−3 − 2 = −5
3 − 2 = 1
Now we have −5 ≤ x ≤ 1.
So x is between −5 and 1: −5 ≤ x ≤ 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.
interval: [−5, 1]
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.
x = −1: 3|−1 + 2| − 4 = 3|1| − 4 = 3(1) − 4
= −1
−1 is less than or equal to 5.
True!
So the answer works.
−1 ≤ 5
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.
▶ Watch it solved
