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

Step by step

The question

Question: solve |x + 1| ≥ 4.

|x + 1| ≥ 4

1 Draw the distance

2 Write it without bars

So x + 1 ≤ −4 or x + 1 ≥ 4.

x + 1 ≤ −4 or x + 1 ≥ 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

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.

x ~~+ 1 − 1~~ ≤ −4 − 1 or x ~~+ 1 − 1~~ ≥ 4 − 1

+ 1 and − 1 make zero, in both.

−4 minus 1 is −5.

4 minus 1 is 3.

−4 − 1 = −5, 4 − 1 = 3

So x ≤ −5 or x ≥ 3.

x ≤ −5 or x ≥ 3

4 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: (−∞, −5] ∪ [3, ∞)

5 Check

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

First −6.

x = −6: |−6 + 1| = |−5| = 5

5 is greater than or equal to 4.

True!

5 ≥ 4

Now 4, from the right part.

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

5 is greater than or equal to 4.

True!

Both parts work!

5 ≥ 4

Let's look back.

First: draw the distance.

Then: write it without bars.

Then: remove the + 1.

Then: show it on a number line.

Last: check.

Answer