Solve |2x + 5| >= 1 Step by Step | Absolute value inequalities

Step by step

The question

Question: solve |2x + 5| ≥ 1.

|2x + 5| ≥ 1

1 Draw the distance

2 Write it without bars

So 2x + 5 ≤ −1 or 2x + 5 ≥ 1.

2x + 5 ≤ −1 or 2x + 5 ≥ 1

"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 + 5

We want x by itself.

The + 5 is in the way.

In each one, we take away 5 on both sides.

2x ~~+ 5 − 5~~ ≤ −1 − 5 or 2x ~~+ 5 − 5~~ ≥ 1 − 5

+ 5 and − 5 make zero, in both.

−1 minus 5 is −6.

1 minus 5 is −4.

−1 − 5 = −6, 1 − 5 = −4

Now we have 2x ≤ −6 or 2x ≥ −4.

2x ≤ −6 or 2x ≥ −4

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.

2 is a positive number, so the sign stays the same.

2x2 ≤ −62 or 2x2 ≥ −42

2 divided by 2 is 1.

So just x is left.

−6 divided by 2 is −3.

−4 divided by 2 is −2.

−62 = −3, −42 = −2

So x ≤ −3 or x ≥ −2.

x ≤ −3 or x ≥ −2

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: (−∞, −3] ∪ [−2, ∞)

6 Check

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

First −4.

x = −4: |2(−4) + 5| = |−8 + 5| = |−3| = 3

3 is greater than or equal to 1.

True!

3 ≥ 1

Now −1, from the right part.

x = −1: |2(−1) + 5| = |−2 + 5| = |3| = 3

3 is greater than or equal to 1.

True!

Both parts work!

3 ≥ 1

Let's look back.

First: draw the distance.

Then: write it without bars.

Then: remove the + 5.

Then: remove the 2.

Then: show it on a number line.

Last: check.

Answer