Chapter 6

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. $$\frac{1}{x}+\frac{1}{x^{2}}=\frac{x+2}{x^{2}}$$

Perform the indicated operations. $$\frac{x^{3}+1}{x^{2}-1} \cdot \frac{3 x-3}{x^{3}-x^{2}+x}$$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{3}{4}=\frac{12}{?}$$

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. $$\frac{1}{x}+\frac{1}{2 x}=\frac{x+2}{2 x}$$

Perform the indicated operations. $$\frac{2 h^{2}-5 h-3}{5 h^{2}-4 h-1} \div \frac{2 h^{2}+7 h+3}{h^{2}+2 h-3}$$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{3}{a}=\frac{?}{a^{2}}$$

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. $$\frac{1}{x}+\frac{1}{x^{2}}=\frac{6}{x^{3}}$$

Perform the indicated operations. $$\frac{9 w^{2}-64}{3 w^{2}-5 w-8} \cdot \frac{5 w^{2}+3 w-2}{25 w^{2}-4}$$

For each pair of polynomials, use division to determine whether the first polynomial is a factor of the second. Use synthetic division when possible. If the first polynomial is a factor, then factor the second polynomial. See Example 7. $$x-1, x^{3}+3 x^{2}-5 x$$

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

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

Perform the indicated operations. $$\frac{9 a-3}{1-9 a^{2}} \cdot \frac{9 a^{2}+6 a+1}{6}$$

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

Perform the indicated operations. $$\frac{5-10 k}{k^{2}-2 k} \div \frac{2 k^{2}+7 k-4}{k^{2}+2 k-8}$$

For each pair of polynomials, use division to determine whether the first polynomial is a factor of the second. Use synthetic division when possible. If the first polynomial is a factor, then factor the second polynomial. See Example 7. $$x+8, x^{2}+3 x-40$$

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

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

Perform the indicated operations. $$\frac{k^{2}+2 k m+m^{2}}{k^{2}-2 k m+m^{2}} \cdot \frac{m^{2}+3 m-m k-3 k}{m^{2}+m k+3 m+3 k}$$

For each pair of polynomials, use division to determine whether the first polynomial is a factor of the second. Use synthetic division when possible. If the first polynomial is a factor, then factor the second polynomial. See Example 7. $$w-3, w^{3}-27$$

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

Perform the indicated operations. $$\frac{a^{2}+2 a b+b^{2}}{a c+b c-a d-b d} \div \frac{a c+a d-b c-b d}{c^{2}-d^{2}}$$

For each pair of polynomials, use division to determine whether the first polynomial is a factor of the second. Use synthetic division when possible. If the first polynomial is a factor, then factor the second polynomial. See Example 7. $$w+5, w^{3}+125$$

Solve each problem. Aspect ratio. The aspect ratio for a television screen is the ratio of the width to the height of the screen. For a high definition TV it is 16 to \(9 .\) What is the height of a screen that is 48 inches wide?

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{a}}{y^{2}} \cdot \frac{y^{b+2}}{x^{2 a}}$$

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

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{2}{w-3}=\frac{-2}{?}$$

Solve each problem. Hybrids. The ratio of the number of cars with gasoline engines to hybrids sold at a car dealer is 9 to \(2 .\) If the dealer sells 10 hybrids one month, then how many gasoline cars did the dealer sell?

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{3 a+1}}{y^{2 b-3}} \cdot \frac{y^{3 b+4}}{x^{2 a-1}}$$

$$ \frac{w^{2}-3 w+6}{w-5}+\frac{9-w^{2}}{w-5} $$$$ \frac{2 z^{2}-3 z+6}{z^{2}-1}-\frac{z^{2}-5 z+9}{z^{2}-1} $$

Solve each problem. Maritime losses. The amount paid to an insured party by the American Insurance Company is computed by using the proportion \(\frac{\text { value shipped }}{\text { amount of loss }}=\frac{\text { amount of declared premium }}{\text { amount insured party gets paid }}\) If the value shipped was \(\$ 300,000,\) the amount of loss was \(\$ 250,000,\) and the amount of declared premium was \(\$ 200,000,\) then what amount is paid to the insured party?

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{2 a}+x^{a}-6}{x^{2 a}+6 x^{a}+9} \div \frac{x^{2 a}-4}{x^{2 a}+2 x^{a}-3}$$

Perform the indicated operations. $$ \frac{2 z^{2}-3 z+6}{z^{2}-1}-\frac{z^{2}-5 z+9}{z^{2}-1} $$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{2 x+4}{6}=\frac{?}{3}$$

Perform the indicated operations. Variables in exponents represent integers. $$\frac{w^{2 b}+2 w^{b}-8}{w^{2 b}+3 w^{b}-4} \div \frac{w^{2 b}-w^{b}-2}{w^{2 b}-1}$$

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

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{2 x-3}{4 x-6}=\frac{1}{?}$$

Solve each problem. Capture-recapture method. To estimate the size of the grizzly bear population in a national park, rangers tagged and released 12 bears. Later it was observed that in 23 sightings of grizzly bears, only two had been tagged. Assuming the proportion of tagged bears in the later sightings is the same as the proportion of tagged bears in the population, estimate the number of bears in the population.

Perform the indicated operations. Variables in exponents represent integers. $$\frac{m^{k} v^{k}+3 v^{k}-2 m^{k}-6}{m^{2 k}-9} \cdot \frac{m^{2 k}-2 m^{k}-3}{v^{k} m^{k}-2 m^{k}+2 v^{k}-4}$$

Perform the indicated operations. $$ \frac{x-1}{2 x^{2}+3 x+1}-\frac{x+1}{2 x^{2}-x-1} $$

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

Solve each problem. Please rewind. In a sample of 24 returned videotapes, it was found that only 3 were rewound as requested. If 872 videos are returned in a day, then how many of them would you expect to find that are not rewound?

Perform the indicated operations. Variables in exponents represent integers. $$\frac{m^{3 k}-1}{m^{3 k}+1} \cdot \frac{m^{2 k+1}-m^{k+1}+m}{m^{3 k}+m^{2 k}+m^{k}}$$

Perform the indicated operations. $$ \frac{2 x+1}{6 x^{2}-5 x+1}+\frac{2 x-1}{6 x^{2}+x-1} $$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{x-3}{x^{2}-9}=\frac{1}{?}$$

Solve each problem. Pleasing painting. The ancient Greeks often used the ratio of length to width for a rectangle as 7 to 6 to give the rectangle a pleasing shape. If the length of a pleasantly shaped Greek painting is 22 centimeters (cm) longer than its width, then what are its length and width?

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

Perform the indicated operations. $$ \frac{2 x+1}{6 x^{2}-5 x+1}+\frac{2 x-1}{6 x^{2}+x-1} $$

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

Solve each problem. Pickups and cars. The ratio of pickups to cars sold at a dealership is 2 to \(3 .\) If the dealership sold 142 more cars than pickups in \(2006,\) then how many of each did it sell?

Perform the indicated operations. $$ \frac{(a b)^{2}}{(a+b)^{2}} \cdot \frac{(a+b)^{3}}{(a b)^{3}} $$