Explore: Complex Fractions Solutions

Complex Fractions Solutions

A complex fraction is a fraction that contains additional fractions, or term(s) with negative exponents, in the numerator and/or denominator. 
With a complex fraction, it is important to remember that the fraction bar is a division symbol. 
The problem can be rewritten horizontally using the division symbol (÷). 
Before dividing, simplify any addition or subtraction in the numerator and denominator.

Example 4

Simplify.

$\frac{2 + \frac{5}{x}}{\frac{2}{x} - 7}$

Plan
Simplify the numerator (dividend)

Simplify the denominator (divisor)

Write as a horizontal fraction using ÷

Take the reciprocal of the second fraction

Simplify

Numerator:

$2 + \frac{5}{x} \begin{matrix} LCD (1, x) = x \end{matrix} \frac{2 x}{x} + \frac{5}{x} \frac{2 x + 5}{x}, x \neq 0$

Denominator:

$\frac{2}{x} - 7 \begin{matrix} L C D (x, 1) = x \end{matrix} \frac{2}{x} - \frac{7 x}{x} \frac{- 7 x + 2}{x}, x \neq 0$

$\frac{2 x + 5}{x} \div \frac{- 7 x + 2}{x} \frac{2 x + 5}{x} \cdot \frac{x}{- 7 x + 2} \frac{2 x + 5}{- 7 x + 2}, x \neq 0, \frac{2}{7}$

Note

Explain

Simplify the numerator and identify excluded values
Simplify the denominator and identify excluded values
Write expressions in standard form
Write simplified numerator and denominator horizontally
Take the reciprocal of the rational expression after the division symbol
Simplify

Example 5

Simplify.

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

Numerator:

$1 - \frac{2}{x} - \frac{3}{x^{2}} \begin{matrix} LCD (1, x, x^{2}) = x^{2} \end{matrix} \frac{x^{2}}{x^{2}} - \frac{2 x}{x^{2}} + \frac{3}{x^{2}} = \frac{x^{2} - 2 x + 3}{x^{2}} \frac{(x - 3) (x + 1)}{x^{2}}, x \neq 0$

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

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

$\frac{4 {(x + 1)}^{2}}{x^{2} (x + 4)}, x \neq - 4, - 1, 0, 3$

Denominator:

$\frac{x}{4} - \frac{3}{x + 1} \begin{matrix} LCD (4, (x + 1)) = 4 (x + 1) \end{matrix} \frac{x (x + 1)}{4 (x + 1)} - \frac{4 (3)}{4 (x + 1)} = \frac{x^{2} + x - 12}{4 (x + 1)} \frac{(x + 4) (x - 3)}{4 (x + 1)}, x \neq - 1$

Example 6

Simplify.

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

Note

The expression raised to the negative first power is the divisor of the complex fraction. Once you find the LCD, take the reciprocal of the divisor to finish simplifying the expression. You do not need to rewrite the problem as shown in this step, but you can if it helps you break down the problem visually.

Implement

$\frac{3}{x - 5} + \frac{1 (x - 5)}{x - 5} \frac{3 + x - 5}{x - 5} \frac{x - 2}{x - 5}, x \neq 5$

$\frac{4 (x + 3)}{x + 3} - \frac{(x - 1)}{x + 3} \frac{4 x + 12 - x + 1}{x + 3} \frac{3 x + 13}{x + 3}, x \neq - 3$

$(\frac{x - 2}{x - 5}) \div (\frac{3 x + 13}{x + 3})$

$(\frac{x - 2}{x - 5}) (\frac{x + 3}{3 x + 13})$

$\frac{(x - 2) (x + 3)}{(x - 5) (3 x + 13)}, x \neq - 3, - \frac{13}{3}, 5$

Explain

Simplify the first group of terms
Simplify the second group of terms
Combine rational expressions
Take the reciprocal of expression after division symbol
Write as one rational expression

Note

Simplified solutions are in factored form in this curriculum.