Solve |x/2 - 1| > 3 Step by Step | Absolute value inequalities

Step by step

The question

Question: solve |x/2 − 1| > 3.

|x2 − 1| > 3

1 Draw the distance

2 Write it without bars

So x/2 − 1 < −3 or x/2 − 1 > 3.

x2 − 1 < −3 or x2 − 1 > 3

"Or" means: x can make one of them true, or the other.

We solve both, side by side.

or: one of them is true

3 Remove the − 1

We want x by itself.

The − 1 is in the way.

In each one, we add 1 on both sides.

x2 ~~− 1 + 1~~ < −3 + 1 or x2 ~~− 1 + 1~~ > 3 + 1

− 1 and + 1 make zero, in both.

−3 plus 1 is −2.

3 plus 1 is 4.

−3 + 1 = −2, 3 + 1 = 4

Now we have x/2 < −2 or x/2 > 4.

x2 < −2 or x2 > 4

4 Remove the fraction

The bottom number is 2.

Times 2 will remove it.

We multiply both sides by 2, in each one.

2 is a positive number, so the signs stay the same.

~~2 ×~~ x2 < 2 × (−2) or ~~2 ×~~ x2 > 2 × 4

2/2 is 1.

The fraction is gone, in both.

2 times −2 is −4.

2 times 4 is 8.

2 × (−2) = −4, 2 × 4 = 8

So x < −4 or x > 8.

x < −4 or x > 8

5 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.

∪ means or.

interval: (−∞, −4) ∪ (8, ∞)

6 Check

Let's test one number from each part, in the question.

First −6.

x = −6: |−62 − 1| = |−3 − 1| = |−4| = 4

4 is greater than 3.

True!

4 > 3

Now 10, from the right part.

x = 10: |102 − 1| = |5 − 1| = |4| = 4

4 is greater than 3.

True!

Both parts work!

4 > 3

Let's look back.

First: draw the distance.

Then: write it without bars.

Then: remove the − 1.

Then: remove the fraction.

Then: show it on a number line.

Last: check.

Answer