Solve (x + 5)/(x - 1) > 3 Step by Step

Step by step

The question

Question: solve (x + 5)/(x − 1) > 3.

x + 5x − 1 > 3

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 − 1.

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 3 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.

x + 5x − 1 − 3 > 3 − 3

Now we have (x + 5)/(x − 1) − 3 > 0.

x + 5x − 1 − 3 > 0

2 Make one fraction

To take away, both need the same bottom.

3 is 3 times (x − 1), over (x − 1).

3 = 3(x − 1)x − 1

Same bottom, so we put the tops together.

x + 5 − 3(x − 1)x − 1 > 0

We open the bracket: minus 3 times each part.

x + 5 − 3x + 3x − 1 > 0

Like terms together: the top is −2x + 8.

Now we have (−2x + 8)/(x − 1) > 0.

−2x + 8x − 1 > 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 each bracket 0.

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

−2x + 8 = 0 x − 1 = 0

In −2x + 8 = 0, we take away 8 on both sides.

In x − 1 = 0, we add 1 on both sides.

−2x + 8 − 8 = 0 − 8 x − 1 + 1 = 0 + 1

Now we have −2x = −8 and x = 1.

2x = −8 x = 1

To remove times −2, we divide both sides by −2.

The critical points are 4 and 1.

x = 4 x = 1

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 1 and 4 split the number line into 3 parts.

At 1 the bottom is 0.

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

✗

At 4, −2x + 8 is 0.

So the fraction is 0 there.

Left of 1, we test x = 0.

−2x + 8 is 8 and x − 1 is −1.

−1

Plus over minus is minus.

Between 1 and 4, we test x = 2.

−2x + 8 is 4 and x − 1 is 1.

Plus over plus is plus.

Right of 4, we test x = 5.

−2x + 8 is −2 and x − 1 is 4.

−2

Minus over plus is minus.

5 Read the answer

Greater than 0 means positive.

So we want the parts with a plus.

So 1 < x < 4.

1 < x < 4

> 0 → the + parts

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.

interval: (1, 4)

7 Check

Let's check: a number from the answer, 2, put into the question.

x = 2: 2 + 52 − 1 = 71 = 7

7 is greater than 3.

True!

7 > 3 true

And a number outside the answer, 0.

x = 0: 0 + 50 − 1 = 5−1 = −5

−5 is not greater than 3.

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

−5 > 3 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