Simplify (2 + 3/x)/(4 - 1/x) Step by Step

Step by step

The question

Question: simplify this fraction.

On top: 2 + 3 over x.

At the bottom: 4 − 1 over x.

A fraction with fractions inside is called a complex fraction.

2 + 3x4 − 1x

1 Find the common bottom

The small fraction has the bottom x.

The smallest thing they all go into is x.

small bottoms: x

common bottom: x

Careful!

We multiply EVERY part by x, also the part with no small bottom.

Leaving it out is the usual mistake.

Careful: every part times x, the whole numbers too

2 Times x on top and bottom

x/x is 1.

Times 1 does not change the value.

xx = 1

We multiply the top and the bottom by x.

Then the small fractions go away.

= x × (2 + 3x)x × (4 − 1x)

3 Multiply every part

Every part times x.

x times 2 is 2x.

x times 3/x: the bottom x goes into x, so we get 3.

x × 2 = 2x

x × 3x = 3

x × 4 = 4x

x × (−1x) = −1

So we get (2x + 3)/(4x − 1).

No more small fractions!

= 2x + 34x − 1

4 Simplify

Is there a number or an x that goes into every part, top and bottom?

No.

So it is already simple.

So the answer is (2x + 3)/(4x − 1).

nothing goes into every part

One more thing: no bottom can be 0.

So x can't be 0 or 0.25.

x ≠ 0, 0.25

5 Check

Let's check with a number: x = 2.

In the question, the long way: the top is 7/2, the bottom is 7/2.

x = 2: top 2 + 32 = 72, bottom 4 − 12 = 72

Top divided by bottom: 1.

question: 72 ÷ 72 = 1

Our answer with x = 2 gives 1 too.

It works!

answer: 2(2) + 34(2) − 1 = 1

Let's look back.

First: find the common bottom.

Then: times x on top and bottom.

Then: multiply every part.

Then: simplify.

Last: check.

Answer