Chapter 6

Use a graphing calculator to graph each rational function. From the graph, determine any vertical asymptotes. See Using Your Calculator: Graphing Rational Functions. $$ f(x)=\frac{x}{x-2} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{a-3}{a+1}=\frac{a-6}{a+5}\)

Simplify each complex fraction. $$ \frac{\frac{7}{a-b}}{\frac{b}{a^{3}-b^{3}}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{2}{x-2}+\frac{3}{x+2}-\frac{x-1}{x^{2}-4}$$

Solve each proportion. $$ \frac{-6}{m}=\frac{m}{-6} $$

Perform each operation and simplify, if possible. See Example 9. $$ \frac{x^{2}-x-12}{x^{2}+x-2} \div \frac{x^{2}-6 x+8}{x^{2}-3 x-10} \cdot \frac{x^{2}-3 x+2}{x^{2}-2 x-15} $$

Use a graphing calculator to graph each rational function. From the graph, determine any vertical asymptotes. See Using Your Calculator: Graphing Rational Functions. $$ f(x)=\frac{x+2}{x} $$

Use synthetic division to perform each division. $$ \frac{5 x^{2}+4+6 x^{3}}{x+1} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{21}{x^{2}-4}-\frac{14}{x+2}=\frac{3}{2-x}\)

Simplify each complex fraction. $$ \frac{\frac{4}{c d}}{c^{-1}+d^{-1}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{8}{x^{2}-9}+\frac{2}{x-3}-\frac{6}{x}$$

Use a proportion to solve each problem. Caffeine. Many convenience stores sell super-size 44 -ounce soft drinks in refillable cups. For each of the products listed in the table below, find the amount of caffeine contained in one of the large cups. Round to the nearest milligram. $$ \begin{array}{|l|c|c|} \hline \text { Soft drink } & \begin{array}{c} \text { Milligrams of } \\ \text { caffeine in } \\ \text { 12 ounces } \end{array} & \begin{array}{c} \text { Milligrams of } \\ \text { caffeine in } \\ \text { 44 ounces } \end{array} \\ \hline \text { Mountain Dew } & 55 & ? \\ \hline \text { Pepsi } & 38 & ? \\ \hline \text { Coca-Cola Classic } & 34 & ? \\ \hline \end{array} $$

Perform each division. \(\left(y^{3}-64\right) \div(y-4)\)

Solve equation. If a solution is extraneous, so indicate. \(\frac{-5}{c+2}=\frac{3}{2-c}+\frac{2 c}{c^{2}-4}\)

Simplify each complex fraction. $$ \frac{\frac{y}{x}+x^{-1}}{x^{-1}+\frac{2 x}{y}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{x}{x^{2}-4}-\frac{x}{x+2}+\frac{2}{x}$$

Use a proportion to solve each problem. Telephones. In \(2009,\) the number of mobile telephone lines used by residents of the United Arab Emirates reached a record high of 191 lines per 100 people. If the Emirates' population was about \(4,599,000\) at that time, how many mobile telephone lines were being used?

Perform each division. \(\left(8 w^{3}+1\right) \div(2 w+1)\)

Perform each operation and simplify, if possible. See Example 9. $$ \frac{4 a^{2}-10 a+6}{a^{4}-3 a^{3}} \div \frac{3-2 a}{2 a^{3}} \cdot \frac{a-3}{2 a-2} $$

Use a graphing calculator to graph each rational function. From the graph, determine any vertical asymptotes. See Using Your Calculator: Graphing Rational Functions. $$ f(x)=\frac{x-2}{x^{2}-3 x-4} $$

Use synthetic division to perform each division. $$ \left(x^{2}-5 x+14\right) \div(x+2) $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{x-4}{x-3}-\frac{x-2}{3-x}=x-3\)

Engineering. The stiffness \(k\) of the shaft shown is given by the formula $$ k=\frac{1}{\frac{1}{k_{1}}+\frac{1}{k_{2}}} $$ where \(k_{1}\) and \(k_{2}\) are the individual stiffnesses of each section. Simplify the complex fraction.

Perform the operations and simplify the result when possible. $$\frac{7}{2 b}-\frac{11}{3 a}$$

Perform each division. \(\frac{a^{8}+a^{6}-4 a^{4}+5 a^{2}-3}{a^{4}+2 a^{2}-3}\)

Perform the operations and simplify. $$ \frac{x^{2}-6 x+9}{4-x^{2}} \div \frac{x^{2}-9}{x^{2}-8 x+12} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{2}+x-30}{3 x^{2}-3 x-60} $$

Use synthetic division to perform each division. $$ \left(x^{2}+13 x+42\right) \div(x+6) $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{5}{x+4}+\frac{1}{x+4}=x-1\)

Transportation. If a bus travels a distance \(d_{1}\) at a speed \(s_{1},\) and then travels a distance \(d_{2}\) at a speed \(s_{2},\) the average (mean) speed \(\bar{s}\) is given by the following formula. Simplify the complex fraction. $$ \bar{s}=\frac{d_{1}+d_{2}}{\frac{d_{1}}{s_{1}}+\frac{d_{2}}{s_{2}}} $$

Perform the operations and simplify the result when possible. $$\frac{5}{3 n}-\frac{7}{4 m}$$

Use a proportion to solve each problem. Recommended Dosages. The recommended child's dose of the sedative hydroxine is 0.006 gram per kilogram of body mass. Find the dosage for a 30 -kg child in grams and in milligrams.

Perform each division. \(\frac{2 x^{4}+3 x^{3}+3 x^{2}-5 x-3}{2 x^{2}-x-1}\)

Perform the operations and simplify. $$ \frac{x^{3}+1}{4} \div \frac{x+1}{2} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{4 x^{2}+24 x+32}{16 x^{2}+8 x-48} $$

Use synthetic division to perform each division. Divide \(a^{5}-1\) by \(a-1\)

Solve equation. If a solution is extraneous, so indicate. \(\frac{a+2}{a+1}=\frac{a-4}{a-3}\)

Perform the operations and simplify the result when possible. $$\frac{s+7}{s+3}-\frac{s-3}{s+7}$$

Perform each division. \(\frac{40 x^{3} z^{2}-8 x^{2} z-4 z}{4 x z}\)

Perform the operations and simplify. $$ \frac{2 x^{2}-2 x-12}{x^{2}-4} \cdot \frac{x^{2}-x-2}{x^{3}-9 x} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{a^{2}-4}{a^{3}-8} $$

Use synthetic division to perform each division. Divide \(b^{4}-81\) by \(b-3\)

Solve equation. If a solution is extraneous, so indicate. \(\frac{z+2}{z+8}=\frac{z-3}{z-2}\)

Data Analysis. Use the data in the table to find the average measurement for the three-trial experiment. Express the answer as a simplified rational expression. $$ \begin{array}{|c|c|c|} \hline \text { Trial 1 } & \text { Trial 2 } & \text { Trial 3 } \\ \hline \frac{k}{3} & \frac{k}{5} & \frac{k}{6} \\ \hline \end{array} $$

Perform the operations and simplify the result when possible. $$\frac{t+5}{t-5}-\frac{t-5}{t+5}$$

Use a proportion to solve each problem. Shopping. A recipe for guacamole dip calls for 5 avocados. If they are advertised at 3 for \(\$ 1.98\), what will 5 avocados cost?

Perform each division. \(\frac{22 a^{2} b^{2}-18 a^{2} b-52 a}{2 a b}\)

Perform the operations and simplify. $$ \frac{x^{2}+2 x-35}{12 x^{3}} \cdot \frac{x^{2}+4 x-21}{x^{2}-3 x} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{3}-27}{3 x^{2}-8 x-3} $$

Use synthetic division to perform each division. $$ \frac{-6 c^{5}+14 c^{4}+38 c^{3}+4 c^{2}+25 c-36}{c-4} $$