Steply

Adding and subtracting rational expressions

Algebra 2 · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Factor the bottoms
  2. Find the common bottom
  3. Make the bottoms the same
  4. Add the tops
  5. Open the brackets
  6. Group like terms
  7. Cross out the same
  8. Check

The steps for Simplify x/(x^2 - 4) + 1/(x + 2). Every solution names its steps, so you always know what comes next.

Worked examples

Simplify 3/(x + 2) + 5/(x + 2) Easy

The question

Question: add 3/(x + 2) + 5/(x + 2).

3x + 2 + 5x + 2

1 Add the tops

The bottoms are already the same.

We add the tops.

The bottom stays.

ac + bc = a + bc

Careful!

The bottoms are the same.

Only the tops are added.

The bottom stays.

Careful: not 8x + 2 + x + 2 ✗

Now one fraction: (3 + 5)/(x + 2).

= 3 + 5(x + 2)

On top: 3 + 5 is 8.

3 + 5 = 8

So the top is 8.

= 8(x + 2)

2 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 8/(x + 2).

The bottoms can't be 0.

So x can't be −2.

x ≠ −2

3 Check

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

First in the question.

3/3 plus 5/3 is 8/3.

Question: 31 + 2 + 51 + 2 = 33 + 53 = 83

Now in our answer.

We get 8/3 too.

Answer: 8(1 + 2) = 83

Both are 8/3.

So 8/(x + 2) is right!

Let's look back.

First: add the tops.

Then: cross out the same.

Last: check.

Answer
= 8(x + 2)

▶ Watch it solved

Simplify 3/x - 1/(x + 4) Medium

The question

Question: work out 3/x − 1/(x + 4).

3x − 1x + 4

1 Find the common bottom

The bottoms are x and (x + 4).

bottoms: x and (x + 4)

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

It is x(x + 4).

common bottom = x(x + 4)

Careful!

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

Fractions need the same bottom first.

Careful: not 2−4 ✗

2 Make the bottoms the same

The first fraction is missing (x + 4).

We multiply its top and bottom by (x + 4).

The second fraction is missing x.

We multiply its top and bottom by x.

(x + 4)/(x + 4) is 1, so the value stays the same.

= 3(x + 4)x(x + 4) − x(x + 4)x

3 Take away the tops

Now one fraction: (3(x + 4) − x)/x(x + 4).

= 3(x + 4) − xx(x + 4)

4 Open the brackets

The 3 in front of the bracket multiplies every part inside.

a(b + c) = ab + ac

So we write 3 times each part.

= 3(x) + 3(4) − xx(x + 4)

3 times x is 3x.

3 times 4 is 12.

= 3x + 12 − xx(x + 4)

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.

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.

= 3x − x + 12x(x + 4)

3x − x is 2x.

It is like 3 apples take away 1 apple: that is 2 apples.

3x − x = 2x

So the expression is 2x + 12.

= 2x + 12x(x + 4)

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 + 12)/x(x + 4).

The bottoms can't be 0.

So x can't be −4 or 0.

x ≠ −4, 0

7 Check

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

First in the question.

3 take away 1/5 is 14/5.

Question: 31 − 11 + 4 = 31 − 15 = 145

Now in our answer.

We get 14/5 too.

Answer: 2(1) + 121(1 + 4) = 145

Both are 14/5.

So (2x + 12)/x(x + 4) 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
= 2x + 12x(x + 4)

▶ Watch it solved

Simplify x/(x^2 - 4) + 1/(x + 2) Harder

The question

Question: add x/(x² − 4) + 1/(x + 2).

xx2 − 4 + 1x + 2

1 Factor the bottoms

x² − 4 is a square minus a square.

So it is (x + 2)(x − 2).

x² − 4 = (x + 2)(x − 2)

We write the bottoms as factors.

= x(x + 2)(x − 2) + 1(x + 2)

2 Find the common bottom

The bottoms are (x + 2)(x − 2) and (x + 2).

bottoms: (x + 2)(x − 2) and (x + 2)

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

It is (x + 2)(x − 2).

common bottom = (x + 2)(x − 2)

Careful!

We can't add the tops and add the bottoms.

Fractions need the same bottom first.

Careful: not x + 1x2 − 2 + x ✗

3 Make the bottoms the same

The first fraction already has it.

The second fraction is missing (x − 2).

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

(x − 2)/(x − 2) is 1, so the value stays the same.

= x(x + 2)(x − 2) + (x − 2)(x + 2)(x − 2)

4 Add the tops

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

= x + (x − 2)(x + 2)(x − 2)

5 Open the brackets

A bracket with a plus or nothing in front can just be removed.

( ) away

Now we have x + x − 2.

= x + x − 2(x + 2)(x − 2)

6 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.

Each colour is one kind.

like terms = same letters,
same powers

x + x is 2x.

It is like 1 apple plus 1 apple: that is 2 apples.

x + x = 2x

So the expression is 2x − 2.

= 2x − 2(x + 2)(x − 2)

7 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 − 2)/(x + 2)(x − 2).

The bottoms can't be 0.

So x can't be −2 or 2.

x ≠ −2, 2

8 Check

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

First in the question.

1 over −3 plus 1/3 is 0.

Question: 112 − 4 + 11 + 2 = 1−3 + 13 = 0

Now in our answer.

We get 0 too.

Answer: 2(1) − 2(1 + 2)(1 − 2) = 0−3 = 0

Both are 0.

So (2x − 2)/(x + 2)(x − 2) is right!

Let's look back.

First: factor the bottoms.

Then: find the common bottom.

Then: make the bottoms the same.

Then: add the tops.

Then: open the brackets.

Then: group like terms.

Then: cross out the same.

Last: check.

Answer
= 2x − 2(x + 2)(x − 2)

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. 3/(x + 2) + 5/(x + 2)
    Answer
    = 8(x + 2)
  2. 7/x - 2/x
    Answer
    = 5x
  3. 4/(x - 1) + 1/(x - 1)
    Answer
    = 5(x − 1)
  4. x/(x + 2) + 2/(x + 2)
    Answer
    = 1
  5. 5/(x + 3) - 2/(x + 3)
    Answer
    = 3(x + 3)
  6. 2/x + 3/(x + 1)
    Answer
    = 5x + 2x(x + 1)
  7. 1/x + 1/(x + 2)
    Answer
    = 2x + 2x(x + 2)
  8. 1/(2x) + 1/(3x)
    Answer
    = 56x

Medium

  1. 3/x - 1/(x + 4)
    Answer
    = 2x + 12x(x + 4)
  2. 4/(x - 2) + 1/x
    Answer
    = 5x − 2x(x − 2)
  3. 5/(2x) - 1/(4x)
    Answer
    = 94x
  4. 5/(x - 2) - 3/(x + 1)
    Answer
    = 2x + 11(x − 2)(x + 1)
  5. 2/(x + 1) + 3/(x - 1)
    Answer
    = 5x + 1(x + 1)(x − 1)
  6. 1/(x + 3) - 1/(x - 3)
    Answer
    = −6(x + 3)(x − 3)
  7. x/(x - 3) - 3/(x - 3)
    Answer
    = 1

Harder

  1. x/(x^2 - 4) + 1/(x + 2)
    Answer
    = 2x − 2(x + 2)(x − 2)
  2. 3/(x^2 - 9) + 1/(x + 3)
    Answer
    = x(x + 3)(x − 3)
  3. 2/(x^2 - 1) + 1/(x + 1)
    Answer
    = 1x − 1
  4. 6/(x^2 + 5x + 6) + 2/(x + 2)
    Answer
    = 2x + 12(x + 2)(x + 3)
  5. 4/(x^2 - 4) - 1/(x - 2)
    Answer
    = −1x + 2