Solve |1 - 2x| > 7 Step by Step | Absolute value inequalities

Step by step

The question

Question: solve |1 − 2x| > 7.

|1 − 2x| > 7

1 Draw the distance

2 Write it without bars

So 1 − 2x < −7 or 1 − 2x > 7.

1 − 2x < −7 or 1 − 2x > 7

"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 take away 1 on both sides.

1 − 2x ~~− 1~~ < −7 − 1 or 1 − 2x ~~− 1~~ > 7 − 1

1 and − 1 make zero, in both.

−7 minus 1 is −8.

7 minus 1 is 6.

−7 − 1 = −8, 7 − 1 = 6

Now we have −2x < −8 or −2x > 6.

−2x < −8 or −2x > 6

4 Remove the −2

−2x means −2 times x.

We want just x.

To remove times −2, we divide by −2.

In each one, on both sides.

We divide by a negative number.

That turns the signs around.

−2x−2 > −8−2 or −2x−2 < 6−2

−2 divided by −2 is 1.

So just x is left.

−8 divided by −2 is 4.

6 divided by −2 is −3.

−8−2 = 4, 6−2 = −3

So x > 4 or x < −3.

x > 4 or x < −3

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: (−∞, −3) ∪ (4, ∞)

6 Check

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

First −4.

x = −4: |1 − 2(−4)| = |1 + 8| = |9| = 9

9 is greater than 7.

True!

9 > 7

Now 5, from the right part.

x = 5: |1 − 2(5)| = |1 − 10| = |−9| = 9

9 is greater than 7.

True!

Both parts work!

9 > 7

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.

Answer