Practice 2 Solutions

Complete each sentence with the word that best describes rational expressions (always, sometimes, never).

The restrictions for the expression sometimes include the numerator and denominator of a rational expression.
Closed means that you will always start and end a problem within the same set of terms.
When the product or quotient of rational expressions is found, the result is always a rational expression.

Simplify.

$\frac{1}{x + 3} + \frac{1}{x - 2} - \frac{1}{x - 3}$

$LCD : (x + 3) (x - 3) (x - 2) x \neq \pm 3, 2 Numerator : (x - 3) (x - 2) + (x + 3) (x - 3) - (x + 3) (x - 2) x^{2} - 5 x + 6 + x^{2} - 9 - (x^{2} + x - 6) x^{2} - 5 x + 6 + x^{2} - 9 - x^{2} - x + 6 x^{2} - 6 x + 3$

$\frac{x^{2} - 6 x + 3}{(x + 3) (x - 3) (x - 2)}, x \neq \pm 3, 2$

$\frac{3 x}{2 x + 5} - \frac{x}{x - 6}$

$LCD : (2 x + 5) (x - 6) x \neq - \frac{5}{2}, 6 Numerator : 3 x (x - 6) - x (2 x + 5) 3 x^{2} - 18 x - 2 x^{2} - 5 x x^{2} - 23 x$

$\frac{x^{2} - 23 x}{(2 x + 5) (x - 6)}, x \neq - \frac{5}{2}, 6$

$\frac{y + 3}{y - 2} - \frac{6 y - 7}{y^{2} - 3 y + 2}$

$\frac{y + 3}{y - 2} - \frac{6 y - 7}{(y - 2) (y - 1)} LCD : (y - 2) (y - 1) y \neq 2, 1 Numerator : (y + 3) (y - 1) - (6 y - 7) y^{2} + 2 y - 3 - 6 y + 7 y^{2} - 4 y + 4 (y - 2) (y - 2) Combined : \frac{(y - 2) (y - 2)}{(y - 2) (y - 1)} \frac{(y - 2) (y - 2)}{(y - 2) (y - 1)}$

$\frac{y - 2}{y - 1}, y \neq 1, 2$

$\frac{4}{x - 5} + \frac{1}{x - 1} - \frac{3 x}{x^{2} - 6 x + 5}$

$\frac{4}{x - 5} + \frac{1}{x - 1} - \frac{3 x}{(x - 5) (x - 1)} LCD : (x - 5) (x - 1) x \neq 5, 1 Numerator : 4 (x - 1) + 1 (x - 5) - 3 x 4 x - 4 + x - 5 - 3 x 2 x - 9$

$\frac{2 x - 9}{(x - 5) (x - 1)}, x \neq 1, 5$

$\frac{2}{x^{2} + 3 x + 2} + \frac{2}{2 x^{2} + 3 x + 1}$

$\frac{2}{(x + 1) (x + 2)} + \frac{2}{(x + 1) (2 x + 1)} LCD : (x + 1) (x + 2) (2 x + 1) x \neq - 1, - 2, - \frac{1}{2} Numerator : 2 (2 x + 1) + 2 (x + 2) 4 x + 2 + 2 x + 4 6 x + 6 6 (x + 1) Combined : \frac{6 (x + 1)}{(x + 1) (x + 2) (2 x + 1)} \frac{6 (x + 1)}{(x + 1) (x + 2) (2 x + 1)}$

$\frac{6}{(x + 2) (2 x + 1)}, x \neq - 1, - 2, - \frac{1}{2}$

$\frac{5}{a} + \frac{2 a - 5}{a^{2} - 10 a} + \frac{1}{a^{2} - 100}$

$\frac{5}{a} + \frac{2 a - 5}{a (a - 10)} + \frac{1}{(a + 10) (a - 10)} LCD : a (a + 10) (a - 10) a \neq 0, \pm 10 Numerator : 5 (a + 10) (a - 10) + (2 a - 5) (a + 10) + a 5 (a^{2} - 100) + 2 a^{2} + 15 a - 50 + a 5 a^{2} - 500 + 2 a^{2} + 15 a - 50 + a 7 a^{2} + 16 a - 550$

$\frac{7 a^{2} + 16 a - 550}{a (a + 10) (a - 10)}, a \neq 0, \pm 10$

$\frac{\frac{4}{y}}{2 - \frac{y}{y + 6}}$

$Numerator LCD : y, y \neq 0 Denominator LCD : y + 6, y \neq - 6 \frac{2 (y + 6) - y}{y + 6} \frac{2 y + 12 - y}{y + 6} \frac{y + 12}{y + 6} Combined : \frac{4}{y} \div \frac{y + 12}{y + 6} \frac{4}{y} \cdot \frac{y + 6}{y + 12}$

$\frac{4 (y + 6)}{y (y + 12)}, y \neq - 12, - 6, 0$

$\frac{\frac{a}{b} + \frac{a}{b + 1}}{\frac{b}{a} + \frac{b}{a + 1}}$

$Numerator LCD : b (b + 1) b \neq 0, - 1 \frac{a (b + 1) + a b}{b (b + 1)} \frac{a b + a + a b}{b (b + 1)} \frac{2 a b + a}{b (b + 1)} \frac{a (2 b + 1)}{b (b + 1)}$

$Denominator LCD : a (a + 1) a \neq 0, - 1 \frac{b (a + 1) + a b}{a (a + 1)} \frac{a b + b + a b}{a (a + 1)} \frac{2 a b + b}{a (a + 1)} \frac{b (2 a + 1)}{a (a + 1)}$
$Combined : \frac{a (2 b + 1)}{b (b + 1)} \div \frac{b (2 a + 1)}{a (a + 1)} \frac{a (2 b + 1)}{b (b + 1)} \cdot \frac{a (a + 1)}{b (2 a + 1)}$

$\frac{a^{2} (a + 1) (2 b + 1)}{b^{2} (2 a + 1) (b + 1)}, a \neq - 1, 0; b \neq - 1, - \frac{1}{2}, 0$

$\frac{\frac{x}{y} - \frac{y}{x}}{\frac{y}{x} - \frac{x}{y}}$

$Numerator LCD : x y, x \neq 0, y \neq 0 \frac{x^{2} - y^{2}}{x y} Denominator LCD : x y, x \neq 0, y \neq 0 \frac{x^{2} - y^{2}}{x y} Combined : \frac{x^{2} - y^{2}}{x y} \div \frac{y^{2} - x^{2}}{x y} \frac{x^{2} - y^{2}}{x y} \cdot \frac{x y}{- (x^{2} - y^{2})} \frac{x^{2} - y^{2}}{x y} \cdot \frac{x y}{- (x^{2} - y^{2})}$

–1

$\frac{\frac{2}{x - 4} + \frac{3}{x + 1}}{\frac{5}{x + 1} - \frac{1}{x - 2}}$

$Numerator LCD : (x + 1) (x - 4), x \neq - 1, 4 \frac{2 (x + 1) + 3 (x - 4)}{(x + 1) (x - 4)} \frac{2 x + 2 + 3 x - 12}{(x + 1) (x - 4)} \frac{5 x - 10}{(x + 1) (x - 4)} \frac{5 (x - 2)}{(x + 1) (x - 4)}$

$Denominator LCD : (x + 1) (x - 2), x \neq - 1, 2 \frac{5 (x - 2) - 1 (x + 1)}{(x + 1) (x - 2)} \frac{5 x - 10 - x - 1}{(x + 1) (x - 2)} \frac{4 x - 11}{(x + 1) (x - 2)}$

$Combined : \frac{5 (x - 2)}{(x + 1) (x - 4)} \div \frac{4 x - 11}{(x + 1) (x - 2)} \frac{5 (x - 2)}{(x + 1) (x - 4)} \cdot \frac{(x + 1) (x - 2)}{(4 x - 11)} \frac{5 (x - 2)}{(x + 1) (x - 4)} \cdot \frac{(x + 1) (x - 2)}{(4 x - 11)}$

$\frac{5 {(x - 2)}^{2}}{(x - 4) (4 x - 11)}, x \neq - 1, 2, \frac{11}{4}, 4$

Find the area of the shaded region.

$A_{shaded region} = A_{rectangle} - A_{triangle} A_{shaded region} = (2 x + 5) (x + 3) A_{triangle} = \frac{1}{2} (\frac{x + 2}{x}) (\frac{x}{x + 1}), x \neq 0, - 1 A_{shaded region} = (2 x + 5) (x + 3) - \frac{1}{2} (\frac{x + 2}{x}) (\frac{x}{x + 1}) A_{shaded region} = (2 x + 5) (x + 3) - \frac{1}{2} (\frac{x + 2}{x}) (\frac{x}{x + 1}) A_{shaded region} = 2 x^{2} + 11 x + 15 - \frac{(x + 2)}{2 (x + 1)}, x \neq 0, - 1$ $LCD : 2 (x + 1), x \neq - 1 Numerator : 2 (x + 1) (2 x^{2} + 11 x + 15) - (x + 2) 2 (2 x^{3} + 11 x^{2} + 15 x + 2 x^{2} + 11 x + 15) - x - 2 2 (2 x^{3} + 13 x^{2} + 26 x + 15) - x - 2 4 x^{3} + 26 x^{2} + 52 x + 30 - x - 2 4 x^{3} + 26 x^{2} + 51 x + 28$

$A_{shaded region} = \frac{4 x^{3} + 26 x^{2} + 51 x + 28}{2 (x + 1)}, x \neq 0, - 1$

Find the area of the trapezoid

$A = \frac{1}{2} h (b_{1} + b_{2}) \frac{1}{2} (x^{2} + 8 x + 7) (\frac{3}{x + 7} + \frac{2}{x + 4}) LCD for b_{1} + b_{2} : (x + 7) (x + 4) x \neq - 7, - 4 \frac{1}{2} (x^{2} + 8 x + 7) (\frac{3 (x + 4) + 2 (x + 7)}{(x + 7) (x + 4)}) \frac{1}{2} (x^{2} + 8 x + 7) (\frac{3 x + 12 + 2 x + 14}{(x + 7) (x + 4)}) \frac{1}{2} \cdot (x + 7) (x + 1) (\frac{5 x + 26}{(x + 7) (x + 4)}) \frac{1}{2} \cdot (x + 7) (x + 1) (\frac{5 x + 26}{(x + 7) (x + 4)})$

$A = \frac{(x + 1) (5 x + 26)}{2 (x + 4)}, x \neq - 7, - 4$