Chapter 6

Solve each equation. $$\frac{-3}{2 x}=\frac{1}{-5}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{x-y}{y-x} \cdot \frac{1}{2}$$

Simplify. $$\left(a^{-1}-b^{-1}\right)^{-2}$$

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

Perform the indicated operations. When possible write down only the answer. $$-1\left(\frac{9-x}{2}\right)$$

Use a calculator to evaluate each complex fraction. Round answers to four decimal places. If your calculator does fractions, then also find the fractional answer. $$\frac{\frac{5}{3}-\frac{4}{5}}{\frac{1}{3}-\frac{5}{6}}$$

Solve each equation. $$\frac{x-2}{x-6}-\frac{4}{x}=\frac{24}{x^{2}-6 x}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{-1}{x-1} \cdot \frac{1-x}{2}$$

Use a calculator to evaluate each complex fraction. Round answers to four decimal places. If your calculator does fractions, then also find the fractional answer. $$\frac{\frac{1}{12}+\frac{1}{2}-\frac{3}{4}}{\frac{3}{5}+\frac{5}{6}}$$

Solve each equation. $$\frac{5}{x^{2}-9}+\frac{2}{x+3}=\frac{1}{x-3}$$

Use a calculator to evaluate each complex fraction. Round answers to four decimal places. If your calculator does fractions, then also find the fractional answer. $$\frac{4^{-1}-9^{-1}}{2^{-1}+3^{-1}}$$

Convert each rational expression into an equivalent rational expression that has the indicated denominator. $$\frac{7}{x-1}, \frac{?}{1-x}$$

Solve each equation. $$\frac{1}{x-2}-\frac{2}{x+3}=\frac{11}{x^{2}+x-6}$$

Use a calculator to evaluate each complex fraction. Round answers to four decimal places. If your calculator does fractions, then also find the fractional answer. $$\frac{2^{-1}+3^{-1}-6^{-1}}{3^{-1}-5^{-1}+4^{-1}}$$

Solve each equation. $$\frac{9}{x^{3}-1}-\frac{1}{x-1}=\frac{2}{x^{2}+x+1}$$

Clarksville has three elementary schools. North side has one-half as many students as Central, and South side has two-thirds as many students as Central. One-third of the students at North side are African-American, three- fourths of the students at Central are African-American, and one-sixth of the students at South side are African-American. What percent of the city’s elementary students are African-American?

Solve each equation. $$\frac{x+4}{x^{3}+8}+\frac{x+2}{x^{2}-2 x+4}=\frac{11}{2 x+4}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{x+3}{\frac{1}{3}}$$

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{4}{x}=\frac{3}{4}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{\frac{3 x}{5}}{y}$$

Mary drove from Clarksville to Leesville at 45 miles per hour (mph). At Leesville she discovered that she had forgotten her purse. She immediately returned to Clarksville at 55 mph. What was her average speed for the entire trip? (The answer is not 50 mph.)

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{5}{h}=\frac{h}{5}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{\frac{b^{2}-4 a}{2}}{a}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{\frac{3 a}{5 b}}{2}$$

Find the indicated value for each given rational expression, if possible. $$R(x)=\frac{3 x-5}{x+4}, R(3)$$

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{5}{h}-\frac{h}{5}$$

Find the indicated value for each given rational expression, if possible. $$T(x)=\frac{5-x}{x-5}, T(-9)$$

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{2}{x}-\frac{3}{4}=\frac{1}{2}$$

Perform the indicated operations. $$\frac{3 x^{2}+13 x-10}{x} \cdot \frac{x^{3}}{9 x^{2}-4} \cdot \frac{7 x-35}{x^{2}-25}$$

Write a step-by-step strategy for simplifying complex fractions with negative exponents. Have a classmate use your strategy to simplify some complex fractions from Exercises 41–60.

Find the indicated value for each given rational expression, if possible. $$H(y)=\frac{y^{2}-5}{3 y-4}, H(-2)$$

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{1}{2 x}-\frac{5}{3 x}=\frac{1}{4}$$

Perform the indicated operations. $$\frac{x^{2}+5 x+6}{x} \cdot \frac{x^{2}}{3 x+6} \cdot \frac{9}{x^{2}-4}$$

a) Find the exact value of each expression. i) \(\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}\) ii) \(\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{3}}}}}\) b) Explain why in each case the exact value must be less than 1.

Find the indicated value for each given rational expression, if possible. $$G(a)=\frac{3-5 a}{2 a+7}, G(5)$$

Perform the indicated operations. $$\frac{\left(a^{2} b^{3} c\right)^{2}}{\left(-2 a b^{2} c\right)^{3}} \cdot \frac{\left(a^{3} b^{2} c\right)^{3}}{(a b c)^{4}}$$

Work with a group to simplify the complex fraction. For what values of \(x\) is the complex fraction undefined? $$\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}}$$

Find the indicated value for each given rational expression, if possible. $$W(b)=\frac{4 b^{3}-1}{b^{2}-b-6}, W(-2)$$

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{1}{2 x}-\frac{5}{3 x}+\frac{1}{4}$$

Perform the indicated operations. $$\frac{\left(-w y^{2}\right)^{3}}{3 w^{2} y} \cdot \frac{(2 w y)^{2}}{4 w y^{3}}$$

Find the indicated value for each given rational expression, if possible. $$N(x)=\frac{x+3}{x^{3}-2 x^{2}-2 x-3}, N(3)$$

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{2 x}=\frac{3}{2 x}$$

Perform the indicated operations. $$\frac{(2 m n)^{3}}{6 m n^{2}} \div \frac{2 m^{2} n^{3}}{\left(m^{2} n\right)^{4}}$$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{1}{3}=\frac{?}{21}$$

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{x^{2}}=\frac{x+1}{x^{2}}$$

Perform the indicated operations. $$\frac{(r t)^{3}}{r t^{4}} \div \frac{\left(r t^{2}\right)^{3}}{r^{2} t^{3}}$$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$4=\frac{?}{3}$$

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{2 x}=\frac{5}{4 x}$$

Perform the indicated operations. $$\frac{2 x^{2}+7 x-15}{4 x^{2}-100} \cdot \frac{2 x^{2}-9 x-5}{4 x^{2}-1}$$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$5=\frac{10}{?}$$