Solve x/(x + 3) >= 1 Step by Step | Rational inequalities

Step by step

The question

Question: solve x/(x + 3) ≥ 1.

xx + 3 ≥ 1

There is an x at the bottom.

The bottom can never be 0.

the bottom can never be 0

1 Get 0 on one side

Careful!

Don't times both sides by x + 3.

It can be negative, and then the sign would turn.

So we get 0 on one side instead.

Careful: never times both sides by the bottom

We take away 1 on both sides.

We do it on both sides.

Why both sides?

The sign compares the two sides.

If we change only one side, the answer can change.

xx + 3 − 1 ≥ 1 − 1

Now we have x/(x + 3) − 1 ≥ 0.

xx + 3 − 1 ≥ 0

2 Make one fraction

To take away, both need the same bottom.

1 is 1 times (x + 3), over (x + 3).

1 = 1(x + 3)x + 3

Same bottom, so we put the tops together.

x − 1(x + 3)x + 3 ≥ 0

We open the bracket: minus 1 times each part.

x − x − 3x + 3 ≥ 0

Like terms together: the top is −3.

Now we have −3/(x + 3) ≥ 0.

−3x + 3 ≥ 0

3 Find the critical points

A critical point is an x where the side with x is 0, or has no value.

Only there can its sign change.

critical point: where it is 0 (or ÷ 0)

We make x + 3 0.

The bottom counts too: there the fraction has no value.

x + 3 = 0

In x + 3 = 0, we take away 3 on both sides.

x + 3 − 3 = 0 − 3

The critical points are −3.

x = −3

4 Make a sign table

Between the critical points the sign stays the same.

So one test number tells us the sign of each part.

We draw a table.

The critical points −3 split the number line into 2 parts.

At −3 the bottom is 0.

Dividing by 0 is not allowed: no value, so we write a cross.

✗

Left of −3, we test x = −4.

−4

x + 3 is −1.

−3

−1

Minus over minus is plus.

Right of −3, we test x = 0.

x + 3 is 3.

−3

Minus over plus is minus.

5 Read the answer

Greater than or equal to 0 means positive or 0.

So we want the plus parts, and the 0s.

But where the bottom is 0, the fraction has no value.

So that point is never part of the answer.

So x < −3.

x < −3

≥ 0 → the + parts, the 0

6 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.

interval: (−∞, −3)

7 Check

Let's check: a number from the answer, −4, put into the question.

x = −4: (−4)(−4) + 3 = −4−1 = 4

4 is greater than or equal to 1.

True!

4 ≥ 1 true

And a number outside the answer, 0.

x = 0: 00 + 3 = 03 = 0

0 is not greater than or equal to 1.

False, as it should be: 0 is not in the answer.

0 ≥ 1 false

Let's look back.

First: get 0 on one side.

Then: make one fraction.

Then: find the critical points.

Then: make a sign table.

Then: read the answer.

Then: show it on a number line.

Last: check.

Answer