Solve x/2 + 1 < -1 or x/2 + 1 > 3 Step by Step

Step by step

The question

Question: solve x/2 + 1 < −1 or x/2 + 1 > 3.

Two inequalities, joined by "or".

x2 + 1 < −1 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

1 Remove the + 1

We want x by itself.

The + 1 is in the way.

In each one, we take away 1 on both sides.

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

+ 1 and − 1 make zero, in both.

−1 minus 1 is −2.

3 minus 1 is 2.

−1 − 1 = −2, 3 − 1 = 2

Now we have x/2 < −2 or x/2 > 2.

x2 < −2 or x2 > 2

2 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 × 2

2/2 is 1.

The fraction is gone, in both.

2 times −2 is −4.

2 times 2 is 4.

2 × (−2) = −4, 2 × 2 = 4

So x < −4 or x > 4.

x < −4 or x > 4

3 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) ∪ (4, ∞)

4 Check

Let's test one number from each part.

First −6, from the left part.

x = −6: −62 + 1 = −3 + 1 = −2

−2 is less than −1.

True!

−2 < −1

Now 6, from the right part.

x = 6: 62 + 1 = 3 + 1 = 4

4 is greater than 3.

True!

Both parts work!

4 > 3

Let's look back.

First: remove the + 1.

Then: remove the fraction.

Then: show it on a number line.

Last: check.

Answer