Algebraic Fractions: Practice

A fraction is undefined where its denominator is zero, so set the bottom equal to zero.

$x - 3 = 0 \;\Rightarrow\; x = 3$

So $x = 3$ is excluded.

The factor $x$ is shared by the top and the bottom, and $\tfrac{x}{x} = 1$.

$\frac{2x}{x} = 2 \cdot \frac{x}{x} = 2$

The original is undefined at $x = 0$, so this holds for $x \neq 0$.

Write $x^2 = x \cdot x$ and cancel the shared factor $x$.

$\frac{x^2}{x} = \frac{x \cdot x}{x} = x$

The coefficients $4$ and $10$ share the factor $2$.

$\frac{4x}{10} = \frac{2 \cdot 2x}{2 \cdot 5} = \frac{2x}{5}$

The greatest common factor of $6x^2$ and $9x$ is $3x$.

$\frac{6x^2}{9x} = \frac{3x \cdot 2x}{3x \cdot 3} = \frac{2x}{3}$

The coefficients share $5$, and the single $x$ in the denominator cancels one of the three powers of $x$ on top, leaving $x^2$.

$\frac{10x^3}{5x} = \frac{5x \cdot 2x^2}{5x \cdot 1} = 2x^2$

Factor the numerator as $5(x + 2)$, then cancel the shared $5$.

$\frac{5x + 10}{5} = \frac{5(x + 2)}{5} = x + 2$

The factor $4$ appears on the top and the bottom, so it cancels.

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

Multiply the tops and the bottoms, then cancel the shared $x$.

$\frac{2}{x} \cdot \frac{x}{5} = \frac{2x}{5x} = \frac{2}{5}$

The original is undefined at $x = 0$, so this holds for $x \neq 0$.

To divide, multiply by the reciprocal of the second fraction.

$\frac{x}{2} \div \frac{x}{6} = \frac{x}{2} \cdot \frac{6}{x} = \frac{6x}{2x} = 3$

The original is undefined at $x = 0$, so this holds for $x \neq 0$.

Multiply by the reciprocal of $\tfrac{2}{x}$, which is $\tfrac{x}{2}$.

$\frac{6}{x} \div \frac{2}{x} = \frac{6}{x} \cdot \frac{x}{2} = \frac{6x}{2x} = 3$

The original is undefined at $x = 0$, so this holds for $x \neq 0$.

The denominators match, so add the like terms in the numerator.

$\frac{2x}{7} + \frac{3x}{7} = \frac{2x + 3x}{7} = \frac{5x}{7}$