Simplify 5/(x - 2) - 3/(x + 1) Step by Step

Step by step

The question

Question: work out 5/(x − 2) − 3/(x + 1).

5x − 2 − 3x + 1

1 Find the common bottom

The bottoms are (x − 2) and (x + 1).

bottoms: (x − 2) and (x + 1)

The common bottom must have every factor of both bottoms, each one once.

It is (x − 2)(x + 1).

common bottom = (x − 2)(x + 1)

Careful!

We can't take away the tops and take away the bottoms.

Fractions need the same bottom first.

Careful: not 2−3 ✗

2 Make the bottoms the same

The first fraction is missing (x + 1).

We multiply its top and bottom by (x + 1).

The second fraction is missing (x − 2).

We multiply its top and bottom by (x − 2).

(x + 1)/(x + 1) is 1, so the value stays the same.

= 5(x + 1)(x − 2)(x + 1) − 3(x − 2)(x + 1)(x − 2)

3 Take away the tops

Now one fraction: (5(x + 1) − 3(x − 2)) over (x − 2)(x + 1).

= 5(x + 1) − 3(x − 2)(x − 2)(x + 1)

4 Open the brackets

What is in front of a bracket multiplies every part inside.

a(b + c) = ab + ac

So we write each part times what is in front.

= 5(x) + 5(1) + (−3)(x) + (−3)(−2)(x − 2)(x + 1)

− × − = +

Watch the signs.

5 times x is 5x.

5 times 1 is 5.

(−3) times x is −3x.

(−3) times (−2) is 6.

= 5x + 5 − 3x + 6(x − 2)(x + 1)

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

Plain numbers go together.

Each colour is one kind.

like terms = same letters,

same powers

We move like terms next to each other.

Each term takes its sign with it.

= 5x − 3x + 5 + 6(x − 2)(x + 1)

5x − 3x is 2x.

It is like 5 apples take away 3 apples: that is 2 apples.

5 + 6 is 11.

5x − 3x = 2x

5 + 6 = 11

So the expression is 2x + 11.

= 2x + 11(x − 2)(x + 1)

6 Cross out the same

Is a whole factor on top the same as one on the bottom?

No.

So nothing crosses out.

We leave the bottom as factors.

So the answer is (2x + 11)/(x − 2)(x + 1).

The bottoms can't be 0.

So x can't be −1 or 2.

x ≠ −1, 2

7 Check

Let's check with a number: x = 1.

First in the question.

5 over −1 take away 3/2 is −13/2.

Question: 51 − 2 − 31 + 1 = 5−1 − 32 = −132

Now in our answer.

We get −13/2 too.

Answer: 2(1) + 11(1 − 2)(1 + 1) = 13−2 = −132

Both are −13/2.

So (2x + 11)/(x − 2)(x + 1) is right!

Let's look back.

First: find the common bottom.

Then: make the bottoms the same.

Then: take away the tops.

Then: open the brackets.

Then: group like terms.

Then: cross out the same.

Last: check.

Answer