Practice Problems

1. Simplify fully:

$$\frac{6x + 12}{10x + 20}$$

2. Simplify fully:

$$\frac{2x^2 – 8x}{4 – x}$$

3. Simplify fully:

$$\frac{x^2 + 7x – 18}{12 – 4x – x^2}$$

4. Simplify fully:

$$\frac{x-y}{2x} \times \frac{x^2}{(x-y)^2}$$

5. Simplify fully:

$$\frac{x^2 – 16}{x^2 + 8x + 16}$$

6. Simplify fully:

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

7. Simplify fully:

$$\frac{3x}{x+1} + \frac{3}{x+1}$$

8. Simplify fully:

$$\frac{5}{6x^2} + \frac{4}{3x}$$

9. Simplify fully:

$$\frac{x+1}{x^2 – x – 6} – \frac{2}{x-3}$$

10. Simplify fully:

$$\frac{7}{x-2} + \frac{4}{2-x}$$

Solutions

1.
Factor numerator and denominator:

$$\frac{6x + 12}{10x + 20} = \frac{6(x+2)}{10(x+2)}$$

Cancel $x+2$ (restriction: $x \neq -2$):

$$= \frac{6}{10}, \quad x \neq -2$$

Simplify:

$$= \frac{3}{5}, \quad x \neq -2$$

2.
Factor numerator:

$$2x^2 – 8x = 2x(x – 4)$$

Rewrite denominator:

$$4 – x = -(x – 4)$$

Simplify:

$$\frac{2x(x – 4)}{-(x – 4)} = \frac{2x}{-1}, \quad x \neq 4$$

Final:

$$= -2x, \quad x \neq 4$$

3.
Factor numerator:

$$x^2 + 7x – 18 = (x + 9)(x – 2)$$

Factor denominator by factoring out $-1$:

$$12 – 4x – x^2 = -(x^2 + 4x – 12) = -(x + 6)(x – 2)$$

Cancel $x – 2$ (restriction: $x \neq 2$):

$$= -\frac{x+9}{x+6}, \quad x \neq 2$$

4.
Multiply:

$$\frac{x-y}{2x} \times \frac{x^2}{(x-y)^2} = \frac{(x-y)x^2}{2x(x-y)^2}$$

Cancel one $x$ and one $x-y$:

$$= \frac{x}{2(x-y)}, \quad x \neq 0, \ x \neq y$$

5.
Factor numerator and denominator:

$$x^2 – 16 = (x – 4)(x + 4)$$ $$x^2 + 8x + 16 = (x + 4)^2$$

Cancel $x+4$ (restriction: $x \neq -4$):

$$= \frac{x – 4}{x + 4}, \quad x \neq -4$$

6.
Rewrite as multiplication by the reciprocal:

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

Factor:

$$= \frac{x-2}{x+3} \times \frac{(x-2)(x+2)}{(x+2)(x-1)}$$

Cancel $x+2$:

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

7.
Same denominator:

$$\frac{3x}{x+1} + \frac{3}{x+1} = \frac{3x+3}{x+1} = \frac{3(x+1)}{x+1}$$

Cancel $x+1$ (restriction: $x \neq -1$):

$$= 3, \quad x \neq -1$$

8.
Common denominator is $6x^2$:

$$\frac{5}{6x^2} + \frac{4}{3x} = \frac{5}{6x^2} + \frac{8x}{6x^2}$$ $$= \frac{5 + 8x}{6x^2}, \quad x \neq 0$$

9.
Factor first:

$$\frac{x+1}{x^2 – x – 6} – \frac{2}{x-3} = \frac{x+1}{(x-3)(x+2)} – \frac{2}{x-3}$$

Lowest common denominator is $(x-3)(x+2)$:

$$= \frac{x+1}{(x-3)(x+2)} – \frac{2(x+2)}{(x-3)(x+2)}$$ $$= \frac{x+1 – 2x – 4}{(x-3)(x+2)} = \frac{-x – 3}{(x-3)(x+2)}$$ $$=\frac{-(x+3)}{(x-3)(x+2)},x\mathrm{\ne }3,-2$$

10.
Note $2 – x = -(x – 2)$:

$$\frac{7}{x-2} + \frac{4}{2-x} = \frac{7}{x-2} – \frac{4}{x-2}$$ $$= \frac{3}{x-2}, \quad x \neq 2$$