Chapter 7

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{5}{6}+\frac{1}{3}}{\frac{2}{3}-\frac{1}{12}} $$

For exercises 27-34, evaluate. $$ \frac{5}{8}+\frac{7}{30} $$

For exercises \(5-48\), simplify. $$ \frac{v^{2}}{2 v-14}-\frac{9 v-14}{2 v-14} $$

For exercises 7-32, simplify. $$ \frac{z^{2}-4 z+6}{z^{2}-9} \cdot \frac{z^{2}+7 z+12}{z^{2}-1} $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x}}{\frac{1}{x}+\frac{1}{x+3}} $$

For exercises 27-34, evaluate. $$ \frac{15}{19}-\frac{8}{11} $$

For exercises \(5-48\), simplify. $$ \frac{x^{2}}{x^{2}-x-12}-\frac{2 x+15}{x^{2}-x-12} $$

Fill in the numerator of \(\frac{?}{x^{2}+4 x-32} \cdot \frac{x^{2}+5 x-24}{x^{2}-12 x+27}\) so that the product is \(\frac{x+2}{x-9}\).

$$ \frac{u-8}{u^{2}-64} $$$$ \frac{k^{2}+5 k+6}{k^{2}+7 k+10} $$

The formula \(R=\frac{V}{I}\) represents the relationship of the resistance \(R\), voltage \(V\), and current \(I\) in an electric circuit. Assume that \(V\) is constant. Is the relationship of \(R\) and \(I\) a direct variation or an inverse variation?

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x}}{\frac{1}{x}+\frac{1}{x+2}} $$

For exercises 27-34, evaluate. $$ \frac{12}{13}-\frac{5}{9} $$

For exercises \(5-48\), simplify. $$ \frac{x^{2}}{x^{2}+3 x-28}-\frac{10 x-24}{x^{2}+3 x-28} $$

Fill in the numerator of \(\frac{?}{x^{2}+4 x-32} \cdot \frac{x^{2}+5 x-24}{x^{2}-11 x+24}\) so that the product is \(\frac{x+2}{x-8}\).

For exercises \(35-36, T=\frac{336 \mathrm{gm}}{R}\) represents the relationship of tire diameter, \(T\); gear ratio, \(g\); speed, \(m\); and revolutions of the tire per minute, \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ g \text { and } m \text { are constant; the relationship of } T \text { and } R $$

For exercises 31-40, (a) solve. (b) check. $$ \frac{3}{w-3}+\frac{4}{w}=\frac{w}{w-3} $$

For exercises \(5-48\), simplify. $$ \frac{2 n^{2}}{2 n^{2}-11 n-21}-\frac{-5 n-3}{2 n^{2}-11 n-21} $$

For exercises 35-38, evaluate. $$ \frac{9}{28} \div \frac{3}{4} $$

$$ \frac{v^{2}+9 v+20}{v^{2}+10 v+24} $$$$ \frac{z^{2}-3 z-40}{z^{2}-64} $$

For exercises \(35-36, T=\frac{336 \mathrm{gm}}{R}\) represents the relationship of tire diameter, \(T\); gear ratio, \(g\); speed, \(m\); and revolutions of the tire per minute, \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ g \text { and } R \text { are constant; the relationship of } T \text { and } m \text {. } $$

For exercises \(5-48\), simplify. $$ \frac{2 w^{2}}{2 w^{2}-11 w-6}-\frac{-5 w-2}{2 w^{2}-11 w-6} $$

For exercises 35-38, evaluate. $$ \frac{16}{27} \div \frac{4}{9} $$

For exercises 37-38, \(T=\frac{R}{A}\) represents the relationship of the asset turnover ratio, \(T\); the sales revenue of a company, \(R\); and the total revenues of a company, \(A\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { is constant; the relationship of } A \text { and } T \text {. } $$

For exercises 31-40, (a) solve. (b) check. $$ \frac{n^{2}}{n-9}-\frac{9 n}{n-9}=-9 $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+3}+\frac{1}{x}}{\frac{1}{x+3}} $$

For exercises 35-38, evaluate. $$ 2 \div \frac{1}{2} $$

For exercises 37-38, \(T=\frac{R}{A}\) represents the relationship of the asset turnover ratio, \(T\); the sales revenue of a company, \(R\); and the total revenues of a company, \(A\). Is the relationship of the given variables a direct variation or an inverse variation? $$ A \text { is constant; the relationship of } R \text { and } T \text {. } $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+2}+\frac{1}{x}}{\frac{1}{x+2}} $$

For exercises \(5-48\), simplify. $$ \frac{y^{3}+5 y^{2}}{y^{3}-16 y}-\frac{36 y}{y^{3}-16 y} $$

For exercises 35-38, evaluate. $$ 3 \div \frac{1}{3} $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+1}-\frac{1}{x}}{\frac{1}{x+1}} $$

For exercises \(35-86\), simplify. $$ \frac{5 r}{21}-\frac{3 r}{10} $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+4}-\frac{1}{x}}{\frac{1}{x+4}} $$

For exercises \(35-86\), simplify. $$ \frac{6 w}{25}-\frac{3 w}{10} $$

For exercises 39-82, simplify. $$ \frac{c d}{b^{2}} \div \frac{d^{2}}{b} $$

For exercises 1-66, simplify. $$ \frac{n^{2}-9}{n^{2}+4 n+3} $$

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ V \text { and } T \text { are constant; the relationship of } R \text { and } C \text {. } $$

If both sides of the equation \(\frac{1}{x-1}+\frac{2}{x}=\frac{x}{x-1}\) are multiplied by \(x(x-1)\), the simplified equation is \(1 x+2(x-1)=x^{2}\). Rewriting in standard form and factoring, the equation is \((x-2)(x-1)=0\) and its solutions are \(x=1\) or \(x=2\). Explain why the solution \(x=1\) is extraneous.

For exercises \(5-48\), simplify. $$ \frac{2 a^{2}}{3 a^{3}+24 a^{2}+45 a}-\frac{10 a+48}{3 a^{3}+24 a^{2}+45 a} $$

For exercises 39-82, simplify. $$ x \div \frac{1}{x} $$

For exercises 1-66, simplify. $$ \frac{u}{u^{2}+6 u} $$

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ C \text { and } T \text { are constant; the relationship of } R \text { and } V \text {. } $$

Author Colin Tudge writes in his book The Variety of Life: "For some have argued that works of art should be self-contained and need no extraneous information to be appreciated: no biography, no history, no referents of any kind." Explain the meaning of extraneous in this statement. (Source: Word of the Day, October 27, 2000, Merriam-Webster Online)

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{4}{x-1}+\frac{2}{x+1}}{\frac{3}{x-1}+\frac{1}{x+1}} $$

For exercises \(5-48\), simplify. $$ \frac{2 c^{2}}{3 c^{3}+18 c^{2}+24 c}-\frac{6 c+56}{3 c^{3}+18 c^{2}+24 c} $$

For exercises 39-82, simplify. $$ z \div \frac{1}{z} $$

For exercises 39-82, simplify. $$ \frac{3 a}{5} \div \frac{9}{10 a^{2}} $$

For exercises 1-66, simplify. $$ \frac{k^{2}-3 k-40}{k^{2}-4 k-45} $$

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ V \text { and } C \text { are constant; the relationship of } R \text { and } T \text {. } $$

For exercises 43-58, (a) solve. (b) check. $$ \frac{3}{k}+\frac{7}{18}=\frac{5}{9} $$