# Add and Subtract Rational Expressions

How to add and subtract rational expressions: 2 examples and their solutions.

## Example 1

### Example

### Solution

Find the least common multiple, LCM,

of the denominators:

(LCM) = x(x + 2).

Change the denominator of 3/x,

x,

to the LCM: x(x + 2).

(x + 2) factor is missing.

So multiply (x + 2)

to both of the numerator and the denominator.

3/x

= 3(x + 2)/[x(x + 2)]

Change the denominator of +x/(x + 2),

(x + 2),

to the LCM: x(x + 2).

x factor is missing.

So multiply x

to both of the numerator and the denominator.

+x/(x + 2)

= +[x⋅x]/[x(x + 2)]

So 3/x + x/(x + 2)

= 3(x + 2)/[x(x + 2)] + [x⋅x]/[x(x + 2)].

3(x + 2) = 3x + 6

x⋅x = x^{2}

The denominators of the fractions are the same.

So add the fractions.

Arrange the terms of the numerator

in descending order.

So [x^{2} + 3x + 6]/[x(x + 2)]

is the answer.

## Example 2

### Example

### Solution

Find the LCM of the denominators.

Factor the denominator of the first fraction

x^{2} - 1.

x^{2} - 1

= x^{2} - 1^{2}

= (x + 1)(x - 1)

Factor the Difference of Two Squares: a^{2} - b^{2}

Factor the denominator of the second fraction

x^{2} + x.

x^{2} + x

= x(x + 1)

Arrange the same factors vertically.

Then the LCM of the denominators is

x(x + 1)(x - 1).

x^{2} - 1 = (x + 1)(x - 1)

x^{2} + x = x(x + 1)

So (given) = 2x/[(x + 1)(x - 1)] - 5/[x(x + 1)].

Change the denominator of 2x/[(x + 1)(x - 1)],

(x + 1)(x - 1),

to the LCM: x(x + 1)(x - 1).

x factor is missing.

So multiply x

to both of the numerator and the denominator.

2x/[(x + 1)(x - 1)]

= 2x⋅x/[x(x + 1)(x - 1)]

Change the denominator of -5/[x(x + 1)],

x(x + 1),

to the LCM: x(x + 1)(x - 1).

(x - 1) factor is missing.

So multiply (x - 1)

to both of the numerator and the denominator.

-5/[x(x + 1)]

= -5(x - 1)/[x(x + 1)(x - 1)]

So 2x/[(x + 1)(x - 1)] - 5/[x(x + 1)]

= 2x⋅x/[x(x + 1)(x - 1)] - 5(x - 1)/[x(x + 1)(x - 1)].

2x⋅x = 2x^{2}

5(x - 1) = 5x - 5

The denominators of the fractions are the same.

So subtract the fractions.

2x^{2} - [5x - 5]

= 2x^{2} - 5x + 5

So [2x^{2} - 5x + 5]/[x(x + 1)(x - 1)]

is the answer.