Algebraic Fractions: Additional Practice (Set 1)

The greatest common factor of $15x^2$ and $10x^3$ is $5x^2$.

$\frac{15x^2}{10x^3} = \frac{5x^2 \cdot 3}{5x^2 \cdot 2x} = \frac{3}{2x}$

The greatest common factor of the numerator is $5$.

$\frac{10x + 25}{5} = \frac{5(2x + 5)}{5} = 2x + 5$

Factor $2x + 8 = 2(x + 4)$, then cancel the shared $x + 4$ and the $x$.

$\frac{3x}{x + 4} \cdot \frac{2(x + 4)}{9x} = \frac{6x(x + 4)}{9x(x + 4)} = \frac{6x}{9x} = \frac{2}{3}$

The original is undefined at $x = 0$ and $x = -4$, so this holds for $x \neq 0,\; x \neq -4$.

Cancel the shared $x - 1$ and the $x$.

$\frac{x - 1}{x} \cdot \frac{4x}{x - 1} = \frac{4x(x - 1)}{x(x - 1)} = \frac{4x}{x} = 4$

The original is undefined at $x = 0$ and $x = 1$, so this holds for $x \neq 0,\; x \neq 1$.

Multiply by the reciprocal $\tfrac{x + 5}{x}$; the $x + 5$ cancels.

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

Factor $8x + 8 = 8(x + 1)$, then multiply by the reciprocal $\tfrac{6}{x + 1}$.

$\frac{8(x + 1)}{3} \cdot \frac{6}{x + 1} = \frac{48(x + 1)}{3(x + 1)} = \frac{48}{3} = 16$

The original divisor is zero at $x = -1$, so this holds for $x \neq -1$.

The LCD of $5$ and $2$ is $10$.

$\frac{2x}{5} + \frac{x}{2} = \frac{4x}{10} + \frac{5x}{10} = \frac{9x}{10}$

The LCD is $x^2$; rewrite $\tfrac{4}{x}$ as $\tfrac{4x}{x^2}$.

$\frac{3}{x^2} + \frac{4}{x} = \frac{3}{x^2} + \frac{4x}{x^2} = \frac{4x + 3}{x^2}$

Distribute the minus sign across both terms of $3x - 4$.

$\frac{7x + 2}{x - 1} - \frac{3x - 4}{x - 1} = \frac{7x + 2 - 3x + 4}{x - 1} = \frac{4x + 6}{x - 1}$

The $-4$ becomes $+4$.

Factor the numerator as $3(2x + 3)$, which shares $2x + 3$ with the denominator.

$\frac{6x + 9}{2x + 3} = \frac{3(2x + 3)}{2x + 3} = 3$

The original is undefined at $x = -\tfrac{3}{2}$, so this holds for $x \neq -\tfrac{3}{2}$.

Set the denominator equal to zero and solve.

$3x + 9 = 0 \;\Rightarrow\; x = -3$

So $x = -3$ is excluded.

The LCD of $2x$ and $3x$ is $6x$.

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