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# Adding and Subtracting Rational Expressions – Techniques & Examples

Before jumping into the topic of **adding and subtracting rational expressions**, let’s remind ourselves what rational expressions are.

Rational expressions are expressions of the form f(x) / g(x) in which the numerator or denominator are polynomials or both the numerator and the numerator are polynomials.

A few examples of rational expression are 3/(x – 1), 4/(2x + 3), (-x + 4)/4, (x^{2} + 9x + 2)/(x + 3), (x + 2)/(x + 6), (x^{2 }– x + 5)/x etc.

## Adding and Subtracting Rational Expressions

To add or subtract rational expressions, **we follow the same steps used for adding and subtracting numerical fractions**.

Just like fractions, adding and subtracting rational expressions of the same denominator is performed by the formula given below:

a/c + b/c = (a + b)/c and a/c – b/c = (a – b)/c

If the denominators of rational expressions are different, we apply the following steps for adding and subtracting rational expressions:

- Factor the denominators to find the least common denominator(LCD)
- Multiply each fraction by the LCD and write the resultant expression over the LCD.
- By keeping the LCD, add or subtract the numerators. Remember to enclose the subtracting numerator in parentheses in order to distribute the subtraction sign.
- Factor the LCD and simplify your rational expression to the lowest terms

### How to subtract rational expressions?

Below are a few examples regarding how to subtract the two rational expressions.

*Example 1*

Solve: 4/x+1 – 1/x + 1

__Solution__

Here, the denominators of both fractions are the same, therefore only subtract the numerators by keeping the denominator.

4/x+1 – 1/x + 1 = (4 – 1)/ 4/x + 1

= 3/x + 1

*Example 2*

Solve (5x – 1)/ (x + 8) – (3x + 8)/ (x + 8)

__Solution__

(5x – 1)/ (x + 8) – (3x + 8)/ (x + 8) = [(5x 1) – (3x + 4)]/ (x + 8)

Now remove the parentheses. Remember to distribute the negative sign accordingly.

= 5x – 1 – 3x – 4/ x +8

subtract the like terms to get;

= 2x -5/x + 8

*Example 3*

Subtract (3x/ x^{2} + 3x -10) – (6/ x^{2} + 3x -10)

__Solution__

The denominators are the same, therefore subtract the numerators only.

(3x/ x^{2} + 3x -10) – (6/ x^{2} + 3x -10) = (3x – 6)/ (x^{2} + 3x -10)

Now factor both the numerator and the denominator to get;

⟹ 3(x -2)/ (x -2) (x + 5)

Simplify the fraction by cancelling out common terms in the numerator and denominator

⟹ 3/ (x + 5)

*Example 4*

Solve: 5/ (x – 4) – 3/ (4 – x)

__Solution__

Factor the denominators to get the LCD

5/ (x – 4) – 3/ (4 – x) ⟹ 5/ (x – 4) – 3/ -1(x – 4)

Therefore, the LCD = x – 4

Multiply each fraction by the LCD.

⟹ 5(x -4)/ (x – 4) – 3(x- 4)/ -1(x – 4)

= [5 – (-3)]/ x – 4

= 8/x -4

*Example 5*

Subtract (2/a) – (3/a −5)

__Solution__

The LCD of the fractions = a (a − 5)

Multiply each fraction by the LCD.

a (a − 5) (2/a) – a (a − 5) (3/a −5) = (2a – 10 – 3a)/a (a – 5)

= (-a -10)/ a (a – 5)

*Example 6*

Subtract 4/ (x^{2 }– 9) – 3/ (x^{2} + 6x + 9)

__Solution__

Factor the denominator of each fraction to get the LCD.

4/ (x^{2 }– 9) – 3/ (x^{2} + 6x + 9) ⟹ 4/ (x -3) (x + 3) – 3/ (x + 3) (x + 3)

Therefore, the LCD = (x -3) (x + 3) (x + 3)

Multiply each fraction by LCD to get;

[4(x + 3) – 3(x – 3)]/ (x -3) (x + 3) (x + 3)

Remove the parentheses in the numerator.

⟹ 4x +12 – 3x + 9/ (x -3) (x + 3) (x + 3)

⟹ x + 21/ (x -3) (x + 3) (x + 3)

Since there is nothing to cancel out, distribute the foil for the denominator to get;

= x + 21/ (x -3) (x + 3)^{2}

### How to add rational expressions?

Below are a few examples regarding how to add the two rational expressions.

*Example 7*

Add 6/ (x – 5) + (x + 2)/(x – 5)

__Solution__

6/ (x – 5) + (x + 2)/(x – 5) = (6 + x + 2)/(x -5)

Combine the like terms

= (8 + x)/(x – 5)

*Example 8*

Simplify (x-2)/(x + 1) + 3/x

__Solution__

LCD = x(x + 1)

Multiply each fraction by LCD

⟹ [x(x + 1)(x-2)/(x + 1) + 3x(x + 1)/x]/ x(x + 1)

= [x (x -2) + 3(x + 1)]/ x(x + 1)

Remove the parentheses in the numerator

= x^{2} – 2x + 3x + 3/ x(x + 1)

Combine like terms;

⟹ x^{2} – x + 3/ x(x + 1)

*Example 9*

Add 1 / (x – 2) + 3 / (x + 4).

__Solution__

There is nothing to factor out in the denominators, therefore we write the LCD as (x – 2)(x + 4).

Multiply each fraction by the LCD

⟹ 1(x – 2)(x + 4)/ (x – 2)) + 3(x – 2)(x + 4) / (x + 4)

= [1(x + 4) – 3(x -2)]/ (x + 4) (x – 2)

Now, remove the parentheses in the numerator

x + 4 – 3x + 6/ (x – 2)(x + 4).

Collect like terms in the numerator.

-x + 10/(x – 2)(x + 4).

There is nothing to factor out, so we FOIL for the denominator to get

= -x + 10 / (x^{2} + 2x – 8)

*Practice Questions*

Simplify the following rational expressions:

- (x – 4)/ 3 + 5x/3
- (2x + 5)/(7) – x/7
- (x + 2)/(x – 7) – ( x
^{2 }+ 4x + 13)/ (x^{2}– 4x -21) - 3 + x/(x + 2) – (2/x
^{2}– 4) - 1/(1 + x) – x/(x – 2) + (x
^{2 }+ 2/x^{2 }– x -2) - 1/(x + y) + (3xy/x
^{3 }+ y^{3}) - (1/a) + a/(2a + 4) – 2/(a
^{2 }+ 2a) - 10x/(5x – 2) + (7x – 2)/(5x – 2)
- 8/(y
^{2}– 4y) + 2/y - 6/( x
^{2}– 4) +2/(x^{2 }– 5x + 6)

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