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# Adding and Subtracting Expressions – Methods & Examples

Do you ever feel dazed when you hear about** the addition and subtraction of rational numbers**? If so, don’t worry, because this is your lucky day!

This article will lead you into a **step-by-step tutorial on how to perform addition and subtraction of rational expressions**, but before that, let’s remind ourselves what rational numbers are.

### Rational number

A rational number is a number that is expressed in the form of p/q, where ‘p’ and ‘q’ are integers and q ≠ 0.

In other words, a rational number is simply a fraction where the integer a is the numerator, and integer b is the denominator.

**Example of rational numbers include**: 2/3, 5/8, -3/14, -11/-5, 7/-9, 7/-15 and -6/-11 etc.

### Algebraic expression

An algebraic expression is a mathematical phrase where variables and constants are combined using the operational (+, -, × & ÷) symbols. For example, 10x + 63 and 5x – 3 are examples of algebraic expressions.

### Rational expression

We have learned that rational numbers are expressed in the form of p/q. On the other hand, a rational expression is a fraction in which either the denominator or the numerator is an algebraic expression. The numerator and the denominator are algebraic expressions.

*Examples of rational expression are:*

3/ (x – 3), 2/ (x + 5), (4x – 1)/3, (x^{2} + 7x)/6, (2x + 5)/(x^{2} + 3x -10), (x+3)/(x + 6) etc.

## How to Add Rational Expressions?

A rational expression with like denominators is added the same way as it is done with fractions. In this case, you keep the denominators and add the numerators together.

* *

*Example 1*

Add (1/4x) + (3/4x)

__Solution__

Keep the denominators and add the numerators alone;

1/4x + 3/4x = (1 + 3)/4x

= 4/4x

Simplify the fraction to its lowest terms;

4/4x = 1/x

* *

*Example 2*

Add (x + 6)/5 + (2x + 4)/5

__Solution__

Keeping the denominator, add the numerators;

(x + 6)/5 + (2x + 4)/5 = [(x + 6) + (2x + 4)]/5

= (x + 6 + 2x + 4)/5

Add the like terms and constants together;

= (x + 2x +6 + 4)5

= (3x + 10)/5

* *

*Example 3*

Add 2/ (x + 7) + 8/ (x +7)

__Solution__

Keeping the denominator, add the numerators;

2/ (x + 7) + 8/ (x +7) = (2 + 8)/ (x + 7)

= 10/ (x + 7)

### Adding rational expressions with unlike denominators

*To add rational expression with different denominators, the following steps are followed:*

- Factor out the denominator
- Determine the least common denominator (LCD). This is done by finding the product of different prime factors and the greatest exponent for each factor.
- Rewrite each rational expression with the LCD as the denominator by multiplying each fraction by 1
- Combine the numerators and keep the LCD as the denominator.
- Reduce the resultant rational expression if possible

* *

*Example 4*

Add 6/x + 3/y

__Solution__

Find the LCD of the denominators. In this case, the LCD = xy.

Rewrite each fraction to contain the LCD as the denominator;

(6/x) (y/y) + (3/y) (x/x)

= 6y/xy + 3x /xy

Now combine the numerators by keeping the denominator;

6y/xy + 3x /xy = (6y +3x)/xy

The fraction cannot be simplified therefore, 6/x + 3/y = (6y +3x)/xy

* *

*Example 5*

Add 4/ (x ^{2} – 16) + 3/ (x ^{2} + 8x + 16)

__Solution__

Start solving by factoring each denominator;

x ^{2} – 16 = (x + 4) (x -4),

And x ^{2} + 8x + 16 = (x +4) (x +4)

= (x + 4)^{2}

4/ (x ^{2} – 16) + 3/ (x ^{2} + 8x + 16) = [4/ (x + 4) (x -4)] + 3/ (x + 4)^{2}

Determine the LCD by finding the product of different prime factors and the greatest exponent for each factor. In this case, the LCD = (x – 4) (x + 4) ^{2}

Rewrite each rational with the LCD as the denominator;

= [4/ (x + 4) (x -4)] (x + 4)/ (x + 4) + 3/ (x + 4)^{2}(x – 4) (x -4)

= (4x + 16)/ [(x – 4) (x +4)^{2}] + (3x – 12/ [(x- 4) (x +4)^{2}]

By keeping the denominators, add the numerators;

= (4x +3x + 16 -12)/ [(x- 4) (x +4)^{2}]

= (7x + 4)/ [(x- 4) (x +4)^{2}]

Since the fraction can be simplified further, hence,

4/ (x ^{2} – 16) + 3/ (x ^{2} + 8x + 16) = (7x + 4)/ [(x- 4) (x +4)^{2}]

## How to Subtract Rational Expressions?

We can subtract rational expressions with like denominators by applying similar steps in addition.

*Let’s take a look at some examples:*

* *

*Example 6*

Subtract 4/(x+1) – 1/ (x + 1)

__Solution__

Subtract the numerators by keeping the denominators;

Hence,

4/(x+1) – 1/ (x + 1) = (4- 1)/ / (x + 1)

= 3/x +1

Therefore, 4/(x+1) – 1/ (x + 1) =3/x +1

* *

*Example 7*

Subtract :(4x – 1)/ (x – 3) – (1 + 3x)/ (x – 3).

__Solution__

Keeping the denominator constant, subtract the numerators;

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

Open up the brackets;

= [4x -1 – 1 – 3x]/(x-3) [ consider the PEMDAS]

= [4x – 3x – 1 -1]/x-3

= (x – 2)/ (x -3)

* *

*Example 8*

Subtract (x^{2} + 7x)/ (x – 7) – (10x + 28)/ (x – 7)

__Solution__

(x^{2} + 7x)/ (x – 7) – (10x + 28)/ (x – 7) = (x ^{2} + 7x – 10x -28)/(x-7)

= (x ^{2} -3x – 28)/ (x -7)

### Subtracting rational expression with unlike denominators

Let’s learn this using a few examples below.

*Example 9*

Subtract 2x / (x^{2} – 9) – 1 / (x + 3)

__Solution__

Factor out the denominators;

x^{2} – 9 = (x + 3) (x – 3).

Now rewrite,

2x / (x + 3) (x – 3) – 1 / (x + 3)

Find the lowest common denominator: LCD = (x + 3) (x – 3)/;

Multiply each fraction by the LCD;

2x – (x – 3) / (x + 3) (x – 3), which simplifies to x + 3 / x^{2} – 9

Therefore,

2x / (x^{2} – 9) – 1 / (x + 3) = x + 3 / x^{2} – 9

* *

*Example 10*

Subtract 2/a – 3/a – 5

__Solution__

Find the LCD;

The LCD = a(a−5).

Rewrite the fraction using the LCD;

2/a – 3/a – 5= 2(a – 5)/ [a (a – 5)] – 3a/[a(a−5)]

Subtract the numerators.

= (2a – 10 – 3a)/ [a(a−5)]

= -a -10/ a(a−5)