Chapter 6

Solve equation. \(\frac{1}{n+2}-\frac{2}{n-3}=\frac{-2 n}{n^{2}-n-6}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{4 y}{y-4}-\frac{16}{y-4}$$

Solve each proportion. $$ \frac{r-2}{3}=\frac{r}{5} $$

Perform each division. \(\frac{54 a^{3} y^{2}-18 a^{4} y^{3}}{27 a^{2} y^{2}}\)

Fund-Raising Letters. Working together, two secretaries can stuff the envelopes for a political fund-raising letter in 4 hours. Working alone, it takes the slower worker 6 hours longer to do the job than the faster worker. How long does it take each to do the job alone?

Multiply, and then simplify, if possible. See Example 2. $$ \frac{x^{2}+2 x+1}{9 x^{3}} \cdot \frac{2 x^{2}-2 x}{2 x^{2}-2} $$

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{3 x-1}{x-x^{2}} $$

Solve equation. \(\frac{2 x}{x^{2}+9 x+20}-\frac{3}{x+4}=\frac{2}{x+5}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{3 x}{2 x+2}+\frac{x+4}{2 x+2}$$

Solve each proportion. $$ \frac{x+1}{x-1}=\frac{6}{4} $$

Perform each division. \(\frac{12 x^{2} y^{3}+x^{3} y^{2}}{6 x y}\)

Surveys. It takes one team 9 days less than another to survey \(1,000\) people. If the teams work together, it takes them 20 days to complete such a survey. How long will it take each to do the survey alone?

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{x^{2}+36}{x^{2}-36} $$

Solve equation. \(\frac{x}{8}=\frac{x-12}{3 x-27}-\frac{1}{3}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{3 x}{x^{2}-9}-\frac{9}{x^{2}-9}$$

Solve each proportion. $$ \frac{5}{5 z+3}=\frac{3}{2 z+6} $$

Perform each division. \(\frac{24 x^{6} y^{7}-12 x^{5} y^{12}+36 x y}{-48 x^{2} y^{3}}\)

It takes 1.5 hours less time for a Cisco Systems server to send out a set of email advertisements than it takes a Dell PowerEdge server to send out the same emails. Working together, the servers can complete the emailing in 1.8 hours. How long would it take each server, working alone, to complete the job?

Multiply, and then simplify, if possible. See Example 2. $$ \frac{x^{3}+3 x^{2}-3 x-9}{x} \cdot \frac{1}{x^{2}+3 x} $$

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{x^{2}+3 x+2}{x^{2}-x-56} $$

Simplify each complex fraction. See Example 4. $$ \frac{\frac{1}{a}+\frac{1}{b}}{\frac{a}{b}-\frac{b}{a}} $$

Use synthetic division to perform each division. See Example 2. $$ \frac{4 c^{3}-107 c+37}{c-5} $$

Solve equation. \(\frac{n}{7}=\frac{n-19}{5 n-45}-\frac{1}{5}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{9 x}{x^{2}-1}-\frac{9}{x^{2}-1}$$

Solve each proportion. $$ \frac{9 t+6}{t}=\frac{7}{3} $$

Perform each division. \(\frac{9 x^{4} y^{3}+18 x^{2} y-27 x y^{4}}{-9 x^{3} y^{3}}\)

Newsletters. \(\quad\) An elementary school teacher can assemble and staple the weekly newsletter three times faster than her student aide. Working together, they can assemble and staple the letters in 12 minutes. How long would it take each, working alone, to complete the job?

Multiply, and then simplify, if possible. See Example 2. $$ \frac{m^{2}+7 m^{2}-7 m-49}{m^{3}} \cdot \frac{1}{m^{2}+7 m} $$

Simplify each complex fraction. See Example 4. $$ \frac{\frac{2}{a^{2}}+\frac{1}{a}}{\frac{2}{a}+\frac{1}{a^{2}}} $$

Use synthetic division to perform each division. See Example 2. $$ \frac{4 t^{3}-t-18}{t-2} $$

Solve equation. \(\frac{p-1}{2}+1=\frac{3}{p}\)

Solve each proportion. $$ \frac{x-2}{x}=\frac{x+1}{x+2} $$

Perform each division. \(\frac{x^{2}+5 x+6}{x+3}\)

Detailing a Car. It takes a man 3 hours to wash and wax the family car. If his teenage son helps him, it only takes 1 hour. How long would it take the son, working alone, to wash and wax the car?

Multiply, and then simplify, if possible. See Example 3. $$ \frac{2 x^{2}-x-3}{x^{2}-1} \cdot \frac{x^{2}+x-2}{2 x^{2}+x-6} $$

Simplify each rational expression. $$ \frac{12 a^{3}}{18 a} $$

Simplify each complex fraction. See Example 4. $$ \frac{\frac{3}{y^{2}}-\frac{4}{y}}{\frac{1}{y}+\frac{15}{y^{2}}} $$

Solve equation. \(\frac{b+1}{2}-\frac{3}{2}=\frac{4}{b}\)

Perform each division. \(\frac{x^{2}+10 x+21}{x+7}\)

Solve each proportion. $$ \frac{a}{a+1}=\frac{a+2}{a} $$

Cleanup Crews. It takes one crew 4 hours to clean an auditorium after an event. If a second crew helps, it only takes 1.5 hours. How long would it take the second crew, working alone, to clean the auditorium?

Multiply, and then simplify, if possible. See Example 3. $$ \frac{2 p^{2}-5 p-3}{p^{2}-9} \cdot \frac{2 p^{2}+5 p-3}{2 p^{2}+5 p+2} $$

Simplify each rational expression. $$ \frac{25 b^{4}}{55 b} $$

Solve equation. \(\frac{16}{t+3}+\frac{7}{t-2}=3\)

Solve each proportion. $$ \frac{2}{3 x}=\frac{6 x}{36} $$

Perform each division. \(\frac{x^{2}-10 x+21}{x-7}\)

Oysters. According to the Guinness Book of World Records, the record for opening oysters is 100 in 140 seconds by Mike Racz in Invercargill, New Zealand, on July \(16,1990\). If it would take a novice \(8 \frac{1}{2}\) minutes to perform the same task, how long would it take them working together to open 100 oysters?

Multiply, and then simplify, if possible. See Example 3. $$ \frac{3 t^{2}-t-2}{6 t^{2}-5 t-6} \cdot \frac{4 t^{2}-9}{2 t^{2}+5 t+3} $$

Simplify each rational expression. $$ \frac{15 a^{2}}{25 a^{8}} $$

Solve equation. \(\frac{17}{s-4}-\frac{10}{s+2}=2\)