Solve (2x + 1)/(x - 1) < 1 Step by Step

Step by step

The question

Question: solve (2x + 1)/(x − 1) < 1.

2x + 1x − 1 < 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 − 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 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.

2x + 1x − 1 − 1 < 1 − 1

Now we have (2x + 1)/(x − 1) − 1 < 0.

2x + 1x − 1 − 1 < 0

2 Make one fraction

To take away, both need the same bottom.

1 is 1 times (x − 1), over (x − 1).

1 = 1(x − 1)x − 1

Same bottom, so we put the tops together.

2x + 1 − 1(x − 1)x − 1 < 0

We open the bracket: minus 1 times each part.

2x + 1 − x + 1x − 1 < 0

Like terms together: the top is x + 2.

Now we have (x + 2)/(x − 1) < 0.

x + 2x − 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.

x + 2 = 0 x − 1 = 0

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

In x − 1 = 0, we add 1 on both sides.

x + 2 − 2 = 0 − 2 x − 1 + 1 = 0 + 1

The critical points are −2 and 1.

x = −2 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 −2 and 1 split the number line into 3 parts.

At −2, x + 2 is 0.

So the fraction is 0 there.

At 1 the bottom is 0.

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

✗

Left of −2, we test x = −3.

−3

x + 2 is −1 and x − 1 is −4.

−1

−4

Minus over minus is plus.

Between −2 and 1, we test x = 0.

x + 2 is 2 and x − 1 is −1.

−1

Plus over minus is minus.

Right of 1, we test x = 2.

x + 2 is 4 and x − 1 is 1.

Plus over plus is plus.

5 Read the answer

Less than 0 means negative.

So we want the parts with a minus.

So −2 < x < 1.

−2 < x < 1

< 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: (−2, 1)

7 Check

Let's check: a number from the answer, 0, put into the question.

x = 0: 2(0) + 10 − 1 = 1−1 = −1

−1 is less than 1.

True!

−1 < 1 true

And a number outside the answer, −3.

x = −3: 2(−3) + 1(−3) − 1 = −5−4 = 54

5/4 is not less than 1.

False, as it should be: −3 is not in the answer.

54 < 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