Fractions

Algebra: Adding and subtracting fractions

Fractions

Fractions with variables

Variables can occur in both the #\orange{\text{numerator}}# as well as the #\blue{\text{denominator }}# of fraction. Just like a fraction with numbers, the #\blue{\text{denominator }}# cannot be equal to #0#. Therefore, we cannot enter numbers for which the denominator is equal to #0#. Most of the time, we will not mention this explicitly, but will assume this value is not entered.

Examples

\[\frac{\orange{x+3}}{\blue{x-5}} \]

\[\frac{\orange{x}}{\blue{x^2-1}} \]

Denominator not equal to 0

Since the #\blue{\text{denominator }}# can not be equal to #0#, the following assumptions hold for the variables.

\[\begin{array}{rcl}
\dfrac{\orange{x+3}}{\blue{x-5}}&&x \ne 5 \\ \\
\dfrac{\orange{x}}{\blue{x^2-1}} && x \ne 1 \text{ and } x \ne -1\end{array}\]

Calculate the following fraction for #x=2#:

#{{\left(x-1\right)\cdot \left(x^2+x\right)}\over{x}}#
Simplify the answer to a simplified fraction or integer.

#3#

This result can be found by replacing #x# with #2# everywhere:
\[\displaystyle {{\left(2-1\right)\cdot \left(2^2+2\right)}\over{2}}= 3\tiny.\]

New example