« Multiplying and Dividing Rational Expressions

Rational Expressions with the Same Denominator

Example 1

Simplify the following:

$\frac{{4x}}{{5y}} + \frac{{6x}}{{5y}}$

Solution

These fractions already have a common denominator

1: Write this sum as the numerator over the common denominator:

$\frac{{4x}}{{5y}} + \frac{{6x}}{{5y}} = \frac{{4x + 6x}}{{5y}}$

2: Reduce to lowest terms:

$\frac{{4x}}{{5y}} + \frac{{6x}}{{5y}} = \frac{{4x + 6x}}{{5y}} = \frac{{10x}}{{5y}} = \frac{{2x}}{y}$

Example 2

Simplify the following:

$\frac{{4x - 1}}{{x + 4}} - \frac{{2x - 9}}{{x + 4}}$

Solution

Again, these already have a common denominator

1: Write this sum as the numerator over the common denominator:

$\frac{{4x - 1}}{{x + 4}} - \frac{{2x - 9}}{{x + 4}} = \frac{{(4x - 1) - (2x - 9)}}{{x + 4}}$

2: Reduce to lowest terms:

$$\frac{{4x - 1}}{{x + 4}} - \frac{{2x - 9}}{{x + 4}} = \frac{{(4x - 1) - (2x - 9)}}{{x + 4}} =$$ $$= \frac{{4x - 1 - 2x + 9}}{{x + 4}} =$$ $$= \frac{{2x + 8}}{{x + 4}} =$$ $$= \frac{{2\cancel{{(x + 4)}}}}{{\cancel{{x + 4}}}} = 2$$

Exercise 1: Simplify the following expression

Adding or Subtracting Rational Expressions with Different Denominators

Example 3:

Simplify the following:

$\frac{{5x - 1}}{{{x^2} - 3x + 2}} + \frac{3}{{2x - 4}}$

Solution 3:

1: Factor each denominator completely.

$\frac{{5x - 1}}{{{x^2} - 3x + 2}} + \frac{2}{{2x - 4}} = \frac{{5x - 1}}{{(x - 1)(x - 2)}} + \frac{3}{{2(x - 2)}}$

2: Build the LCD of the denominators.

$LCD = 2(x - 1)(x - 2)$

3: Rewrite each rational expression with the LCD as the denominator.

$$\frac{{5x - 1}}{{{x^2} - 3x + 2}} + \frac{3}{{2x - 4}} = \frac{{5x - 1}}{{(x - 1)(x - 2)}} + \frac{3}{{2(x - 2)}} =$$ $$= \frac{{2(5x - 1)}}{{2(x - 1)(x - 2)}} + \frac{{3(x - 1)}}{{2(x - 1)(x - 2)}}$$

4: Add the numerators.

$$\frac{{5x - 1}}{{{x^2} - 3x + 2}} + \frac{3}{{2x - 4}} = \frac{{5x - 1}}{{(x - 1)(x - 2)}} + \frac{3}{{2(x - 2)}} =$$ $$= \frac{{2(5x - 1)}}{{2(x - 1)(x - 2)}} + \frac{{3(x - 1)}}{{2(x - 1)(x - 2)}} =$$ $$= \frac{{2(5x - 1) + 3(x - 1)}}{{2(x - 1)(x - 2)}} =$$ $$= \frac{{13x - 5}}{{2(x - 1)(x - 2)}}$$

Example 4:

Simplify the following:

$\frac{{5x + 1}}{{{x^2} - 2x - 3}} - \frac{{5x - 3}}{{{x^2} - x - 6}}$

Solution 4:

1: Factor each denominator completely.

$\frac{{5x + 1}}{{{x^2} - 2x - 3}} - \frac{{5x - 3}}{{{x^2} - x - 6}} = \frac{{5x + 1}}{{(x - 3)(x + 1)}} - \frac{{5x - 3}}{{(x - 3)(x + 2)}}$

2: Build the LCD of the denominators.

$LCD = (x - 3)(x + 1)(x + 2)$

3: Rewrite each rational expression with the LCD as the denominator.

$$\frac{{5x + 1}}{{{x^2} - 2x - 3}} - \frac{{5x - 3}}{{{x^2} - x - 6}} = \frac{{5x + 1}}{{(x - 3)(x + 1)}} - \frac{{5x - 3}}{{(x - 3)(x + 2)}} =$$ $$= \frac{{(5x + 1)(x + 2)}}{{(x - 3)(x + 1)(x + 2)}} - \frac{{(5x - 3)(x + 1)}}{{(x - 3)(x + 1)(x + 2)}}$$

4: Subtract the numerators.

$$\frac{{5x + 1}}{{{x^2} - 2x - 3}} - \frac{{5x - 3}}{{{x^2} - x - 6}} = \frac{{5x + 1}}{{(x - 3)(x + 1)}} - \frac{{5x - 3}}{{(x - 3)(x + 2)}} =$$ $$= \frac{{(5x + 1)(x + 2)}}{{(x - 3)(x + 1)(x + 2)}} - \frac{{(5x - 3)(x + 1)}}{{(x - 3)(x + 1)(x + 2)}} =$$ $$= \frac{{(5x + 1)(x + 2) - (5x - 3)(x + 1)}}{{(x - 3)(x + 1)(x + 2)}} =$$ $$= \frac{{(5{x^2} + 10x + x + 2) - (5{x^2} + 5x - 3x - 3)}}{{(x - 3)(x + 1)(x + 2)}} =$$ $$= \frac{{5{x^2} + 10x + x + 2 - 5{x^2} - 5x + 3x + 3)}}{{(x - 3)(x + 1)(x + 2)}} =$$ $$= \frac{{9x + 5}}{{(x - 3)(x + 1)(x + 2)}}$$

Exercise 2: Simplify the following expression