Mixed Expressions

A mixed expression, commonly called a complex fraction , is a fraction where the numerator, the denominator or both the numerator and denominator contain a fraction.

We can use the rules of algebra to manipulate these expressions into rational ones.

First, rewrite the denominator as a single rational expression.

$\begin{array}{l}\frac{\left(\frac{3x}{x+1}\right)}{\left(2+\frac{x}{4x-2}\right)}=\frac{\left(\frac{3x}{x+1}\right)}{\left(\frac{\left(8x-4\right)+x}{4x-2}\right)}\\ =\frac{\left(\frac{3x}{x+1}\right)}{\left(\frac{9x-4}{4x-2}\right)}\end{array}$

Since the fraction line means divide, rewrite the problem as multiplication by the reciprocal .

$\begin{array}{l}=\frac{3x}{x+1}\cdot \frac{4x-2}{9x-4}\\ =\frac{12x^2-6x}{9x^2+5x-4}\end{array}$

This is a rational expression.

A second method we can use is to multiply the numerator and denominator of the original improper fraction by the LCD of the two fractions (which will be another name for $1$ ).

Multiply both numerator and denominator by $2x\left(x+1\right)$ .

$\begin{array}{l}=\frac{\left(\frac{3x}{\cancel{x+1}}\cdot 2x\cancel{\left(x+1\right)}\right)}{\left(\frac{5}{\cancel{2x}}\cdot \cancel{2x}\left(x+1\right)\right)}\\ =\frac{3x\cdot 2x}{5\left(x+1\right)}\\ =\frac{6x^2}{5x+5}\end{array}$