Solve |x + 4| - 1 >= 3 Step by Step | Absolute value inequalities

Step by step

The question

Question: solve |x + 4| − 1 ≥ 3.

|x + 4| − 1 ≥ 3

1 Get the bars alone

We want the bars by themselves.

The − 1 is in the way, so we add 1 on both sides.

|x + 4| ~~− 1 + 1~~ ≥ 3 + 1

− 1 and + 1 make zero.

Cross them out.

3 plus 1 is 4.

|x + 4| ≥ 4

2 Draw the distance

3 Write it without bars

So x + 4 ≤ −4 or x + 4 ≥ 4.

x + 4 ≤ −4 or x + 4 ≥ 4

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

We solve both, side by side.

or: one of them is true

4 Remove the + 4

We want x by itself.

The + 4 is in the way.

In each one, we take away 4 on both sides.

x ~~+ 4 − 4~~ ≤ −4 − 4 or x ~~+ 4 − 4~~ ≥ 4 − 4

+ 4 and − 4 make zero, in both.

−4 minus 4 is −8.

4 minus 4 is 0.

−4 − 4 = −8, 4 − 4 = 0

So x ≤ −8 or x ≥ 0.

x ≤ −8 or x ≥ 0

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.

∞ means it goes on for ever.

∪ means or.

interval: (−∞, −8] ∪ [0, ∞)

6 Check

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

First −9.

x = −9: |−9 + 4| − 1 = |−5| − 1 = 5 − 1 = 4

4 is greater than or equal to 3.

True!

4 ≥ 3

Now 1, from the right part.

x = 1: |1 + 4| − 1 = |5| − 1 = 5 − 1 = 4

4 is greater than or equal to 3.

True!

Both parts work!

4 ≥ 3

Let's look back.

First: get the bars alone.

Then: draw the distance.

Then: write it without bars.

Then: remove the + 4.

Then: show it on a number line.

Last: check.

Answer