Practice 1 Solutions

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

The sum of rational expressions is always a rational expression.
The denominator of a rational expression should never be undefined.
Rational expressions are always closed under addition, subtraction, multiplication, and division.

Simplify.

Note

See Lesson 3 More to Explore for ways to use technology to check answers.

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

$LCD : (x + 3) (x + 2) x \neq - 3, - 2 Numerator : 3 x (x + 2) - (x + 1) (x + 3) 3 x^{2} + 6 x - (x^{2} + 4 x + 3) 3 x^{2} + 6 x - x^{2} - 4 x - 3 2 x^{2} + 2 x - 3$

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

Note

Q: What are the restrictions for the denominator?
A: –3, –2

$\frac{y}{y - 1} + \frac{5}{y + 2} - \frac{3}{y^{2} + y - 2}$

$\frac{y}{y - 1} + \frac{5}{y + 2} - \frac{3}{(y - 1) (y + 2)} LCD : (y - 1) (y + 2) y \neq 1, - 2 Numerator : y (y + 2) + 5 (y - 1) - 3 y^{2} + 2 y + 5 y - 5 - 3 y^{2} + 7 y - 8 Combined : \frac{(y + 8) (y - 1)}{(y - 1) (y + 2)} \frac{(y + 8) (y - 1)}{(y - 1) (y + 2)}$

$\frac{y + 8}{y + 2}, y \neq - 2, 1$

Note

If you do not factor the numerator, you will not find the completely simplified answer.

$\frac{7}{m + 5} + \frac{4}{5 - m} + \frac{2 m - 1}{m^{2} - 25}$

$\frac{7}{m + 5} + \frac{4}{- (m - 5)} + \frac{2 m - 1}{(m + 5) (m - 5)} \frac{7}{m + 5} + \frac{- 4}{m - 5} + \frac{2 m - 1}{(m + 5) (m - 5)} LCD : (m + 5) (m - 5) m \neq \pm 5 Numerator : 7 (m - 5) - 4 (m + 5) + 2 m - 1 7 m - 35 - 4 m - 20 + 2 m - 1 5 m - 56$

$\frac{5 m - 56}{(m + 5) (m - 5)}, m \neq \pm 5$

Note

The middle numerator will be negative when –1 is factored from the denominator. Either the numerator or denominator is negative in a negative fraction.

Q: What should be factored from an expression when the variable is negative?

A: –1

Alternate answer: $LCD : (- 1) (m + 5) (m - 5) Numerator : (- 1) (7) (m - 5) + 4 (m + 5) + (- 1) (2 m - 1) - 7 m + 35 - 4 m + 20 - 2 m + 1 - 5 m + 56 \frac{- 5 m + 56}{(- 1) (m + 5) (m - 5)}, m \neq \pm 5$

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

$\frac{2 x + 3}{2 x (2 x + 3)} + \frac{2 x}{2 x + 3} + \frac{3}{2 x} LCD : 2 x (2 x + 3) x \neq 0, - \frac{3}{2} Numerator : 2 x + 3 + 2 x (2 x) + 3 (2 x + 3) 2 x + 3 + 4 x^{2} + 6 x + 9 4 x^{2} + 8 x + 12 Combined : \frac{4 (x^{2} + 2 x + 3)}{2 x (2 x + 3)} \frac{{}^{2}4 (x^{2} + 2 x + 3)}{2 x (2 x + 3)}$

$\frac{2 (x^{2} + 2 x + 3)}{x (2 x + 3)}, x \neq - \frac{3}{2}, 0$

Note

Remember to factor out the GCF of the numerator so you can completely simplify the expression. The numerator will factor, however the solution would not be further simplified, so it is not necessary. If you wrote the solution as $\frac{2 (x + 1) (x + 3)}{x (2 x + 3)}$ you would also be correct.

$\frac{r^{2} - 6 r - 7}{r^{2} - 3 r - 28} + \frac{r^{2} + 4 r + 3}{r^{2} + 3 r + 2}$

$\frac{(r - 7) (r + 1)}{(r - 7) (r + 4)} + \frac{(r + 3) (r + 1)}{(r + 2) (r + 1)} \frac{(r - 7) (r + 1)}{(r - 7) (r + 4)} + \frac{(r + 3) (r + 1)}{(r + 2) (r + 1)} \frac{r + 1}{r + 4} + \frac{r + 3}{r + 2} LCD : (r + 4) (r + 2) r \neq - 4, - 2, - 1, 7 Numerator : (r + 1) (r + 2) + (r + 4) (r + 3) r^{2} + 3 r + 2 + r^{2} + 7 r + 12 2 r^{2} + 10 r + 14$

$\frac{2 (r^{2} + 5 r + 7)}{(r + 4) (r + 2)}, r \neq - 4, - 2, - 1, 7$

Note

(r + 1) is simplified out of the second fraction, so it is not needed for the LCD.

$\frac{2 g}{9 g^{2} - 25} - \frac{3 g + 5}{5 - 3 g} + \frac{g}{3 g + 5}$

$\frac{2 g}{(3 g + 5) (3 g - 5)} - \frac{3 g + 5}{- (3 g - 5)} + \frac{g}{3 g + 5} \frac{2 g}{(3 g + 5) (3 g - 5)} + \frac{3 g + 5}{3 g - 5} + \frac{g}{3 g + 5} LCD : (3 g + 5) (3 g - 5) g \neq \pm \frac{5}{3} Numerator : 2 g + (3 g + 5) (3 g + 5) + g (3 g - 5) 2 g + 9 g^{2} + 30 g + 25 + 3 g^{2} - 5 g 12 g^{2} + 27 g + 25$

$\frac{12 g^{2} + 27 g + 25}{(3 g + 5) (3 g - 5)}, g \neq \pm \frac{5}{3}$

Note

Problems 10–14
Perseverance is important when working with complex fractions. Keep at it! If you get stuck, remember that you can break down problems by the numerator and denominator as shown in the video and notes.

$\frac{\frac{x}{y} + 3}{\frac{x}{y} - 4 y}$

$Numerator LCD : y, y \neq 0 \frac{x}{y} + 3 \frac{x}{y} + \frac{3 y}{y} = \frac{x + 3 y}{y} Denominator LCD : y, y \neq 0 \frac{x}{y} - 4 y \frac{x}{y} - \frac{4 y^{2}}{y} = \frac{x - 4 y^{2}}{y}$
$Combined : \frac{x + 3 y}{y} \div \frac{x - 4 y^{2}}{y} \frac{x + 3 y}{y} \cdot \frac{y}{x - 4 y^{2}} \frac{x + 3 y}{y} \cdot \frac{y}{x - 4 y^{2}}$

$\frac{x + 3 y}{x - 4 y^{2}}, y \neq 0$

$\frac{\frac{2}{a} + \frac{3}{b}}{\frac{6 b + 9 a}{a b}}$

$Numerator LCD : a b, a \neq 0, b \neq 0 \frac{2 b}{a b} + \frac{3 a}{a b} = \frac{2 b + 3 a}{a b} Denominator LCD : a b, a \neq 0, b \neq 0 \frac{3 (2 b + 3 a)}{a b} Combined : \frac{2 b + 3 a}{a b} \div \frac{3 (2 b + 3 a)}{a b} \frac{2 b + 3 a}{a b} \cdot \frac{a b}{3 (2 b + 3 a)} \frac{2 b + 3 a}{a b} \cdot \frac{a b}{3 (2 b + 3 a)}$

$\frac{1}{3}, a \neq 0, b \neq 0$

Note

Problems 12–13
The denominator of the simplified expression does not have excluded values under the set of real numbers.

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

$Numerator LCD : x (x + 1), x \neq 0, - 1 \frac{x + 1}{x (x + 1)} + \frac{2 x}{x (x + 1)} \frac{3 x + 1}{x (x + 1)}$

$Denominator LCD : x (x + 2) (x + 1), x \neq 0, - 2, - 1 \frac{2 x (x + 1)}{x (x + 2) (x + 1)} - \frac{x (x + 2)}{x (x + 2) (x + 1)} + \frac{3 (x + 2) (x + 1)}{x (x + 2) (x + 1)} \frac{2 x (x + 1) - x (x + 2) + 3 (x + 2) (x + 1)}{x (x + 2) (x + 1)} \frac{2 x^{2} + 2 x - x^{2} - 2 x + 3 (x^{2} + 3 x + 2)}{x (x + 2) (x + 1)} \frac{2 x^{2} + 2 x - x^{2} - 2 x + 3 x^{2} + 9 x + 6}{x (x + 2) (x + 1)} \frac{4 x^{2} + 9 x + 6}{x (x + 2) (x + 1)}$
$Combined : \frac{3 x + 1}{x (x + 1)} \div \frac{4 x^{2} + 9 x + 6}{x (x + 2) (x + 1)} \frac{3 x + 1}{x (x + 1)} \cdot \frac{x (x + 2) (x + 1)}{4 x^{2} + 9 x + 6} \frac{3 x + 1}{x (x + 1)} \cdot \frac{x (x + 2) (x + 1)}{4 x^{2} + 9 x + 6}$

$\frac{(3 x + 1) (x + 2)}{4 x^{2} + 9 x + 6}, x \neq - 2, - 1, 0$

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

$Numerator LCD : (x + 3) (x - 3), x \neq \pm 3 \frac{(x + 3)}{(x + 3) (x - 3)} + \frac{5 (x - 3)}{(x + 3) (x - 3)} \frac{x + 3 + 5 (x - 3)}{(x + 3) (x - 3)} \frac{x + 3 + 5 x - 15}{(x + 3) (x - 3)} \frac{6 x - 12}{(x + 3) (x - 3)} \frac{6 (x - 2)}{(x + 3) (x - 3)}$
$Denominator LCD : x (x - 3), x \neq 0, 3 \frac{3 (x - 3)}{x (x - 3)} - \frac{2 x (x)}{x (x - 3)} \frac{3 (x - 3) - 2 x^{2}}{x (x - 3)} \frac{3 x - 9 - 2 x^{2}}{x (x - 3)} \frac{- 2 x^{2} + 3 x - 9}{x (x - 3)}$

$Combined : \frac{6 (x - 2)}{(x + 3) (x - 3)} \div \frac{- 2 x^{2} + 3 x - 9}{x (x - 3)} \frac{6 (x - 2)}{(x + 3) (x - 3)} \cdot \frac{x (x - 3)}{- 2 x^{2} + 3 x - 9} \frac{6 (x - 2)}{(x + 3) (x - 3)} \cdot \frac{x (x - 3)}{- 2 x^{2} + 3 x - 9}$

$\frac{6 x (x - 2)}{(x + 3) (- 2 x^{2} + 3 x - 9)}, x \neq \pm 3, 0$

An electrical circuit with three resistors connected in parallel is shown using the equation: $R_{T} = \frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}}$ . Simplify the expression.

$Denominator LCD : R_{1} R_{2} R_{3}, R_{1} \neq 0, R_{2} \neq 0, R_{3} \neq 0 R_{T} = \frac{1}{\frac{R_{2} R_{3} + R_{1} R_{3} + R_{1} R_{2}}{R_{1} R_{2} R_{3}}} R_{T} = 1 \div \frac{R_{2} R_{3} + R_{1} R_{3} + R_{1} R_{2}}{R_{1} R_{2} R_{3}}$

$R_{T} = \frac{R_{1} R_{2} R_{3}}{R_{2} R_{3} + R_{1} R_{3} + R_{1} R_{2}}, R_{1} \neq 0, R_{2} \neq 0, R_{3} \neq 0$

Note

If you do not like to work with the subscripts for R, you can substitute other variables to simplify and then substitute back the subscript Rs to check your final answer.

Find the total resistance of the electrical circuit if three light bulbs in parallel have the resistance of: $R_{1} = \frac{2}{5} ohms, R_{2} = \frac{5}{3} ohms, and R_{3} = 4 ohms$ .

$R_{T} = \frac{\frac{2}{5} \cdot \frac{5}{3} \cdot \frac{4}{1}}{\frac{5}{3} \cdot \frac{4}{1} + \frac{2}{5} \cdot \frac{4}{1} + \frac{2}{5} \cdot \frac{5}{3}} R_{T} = \frac{\frac{8}{3}}{\frac{20}{3} + \frac{8}{5} + \frac{2}{3}} R_{T} = \frac{\frac{8}{3}}{\frac{100}{15} + \frac{24}{15} + \frac{10}{15}} R_{T} = \frac{\frac{8}{3}}{\frac{134}{15}} R_{T} = \frac{8}{3} \div \frac{134}{15} R_{T} = \frac{8}{3} \cdot \frac{15}{134} = \frac{(4) (2) (5) (3)}{(3) (2) (67)}$

$R_{T} = \frac{20}{67}$