Chapter 7

In Exercises 25–32, graph the function. State the domain and range. $$ y=\frac{x-1}{x+5} $$

Find the sum or difference. \(\frac{x+3}{x^2-25}-\frac{x-1}{x-5}+\frac{3}{x+3}\)

Solve the equation by using the LCD. Check your solution(s). $$\frac{10}{x^2-2 x}+\frac{4}{x}=\frac{5}{x-2}$$

In Exercises 25–32, graph the function. State the domain and range. $$ y=\frac{x+6}{4 x-8} $$

Solve the equation by using the LCD. Check your solution(s). $$\frac{x+1}{x+6}+\frac{1}{x}=\frac{2 x+1}{x+6}$$

Find the quotient. $$ \frac{32 x^3 y}{y^8} \div \frac{y^7}{8 x^4} $$

In Exercises 25–32, graph the function. State the domain and range. $$ h(x)=\frac{8 x+3}{2 x-6} $$

Tell whether the statement is always, sometimes, or never true. Explain. The LCD of two rational expressions will have a degree greater than or equal to that of the denominator with the higher degree.

Solve the equation by using the LCD. Check your solution(s). $$\frac{x+3}{x-3}+\frac{x}{x-5}=\frac{x+5}{x-5}$$

Find the quotient. $$ \frac{2 x y z}{x^3 z^3} \div \frac{6 y^4}{2 x^2 z^2} $$

In Exercises 25–32, graph the function. State the domain and range. $$ f(x)=\frac{-5 x+2}{4 x+5} $$

How would you begin to rewrite the function \(g(x)=\frac{4 x+1}{x+2}\) to obtain the form \(g(x)=\frac{a}{x-h}+k ?\) (A) \(g(x)=\frac{4(x+2)-7}{x+2}\) (B) \(g(x)=\frac{4(x+2)+1}{x+2}\) (C) \(g(x)=\frac{(x+2)+(3 x-1)}{x+2}\) (D) \(g(x)=\frac{4 x+2-1}{x+2}\)

MAKING AN ARGUMENT You have enough money to buy 5 hats for \(\$ 10\) each or 10 hats for \(\$ 5\) each. Your friend says this situation represents inverse variation. Is your friend correct? Explain your reasoning.

Solve the equation by using the LCD. Check your solution(s). $$\frac{5}{x}-2=\frac{2}{x+3}$$

Find the quotient. $$ \frac{x^2-x-6}{2 x^4-6 x^3} \div \frac{x+2}{4 x^3} $$

In Exercises 25–32, graph the function. State the domain and range. $$ g(x)=\frac{6 x-1}{3 x-1} $$

How would you begin to rewrite the function \(g(x)=\frac{x}{x-5}\) to obtain the form \(g(x)=\frac{a}{x-h}+k ?\) (A) \(g(x)=\frac{x(x+5)(x-5)}{x-5}\) (B) \(g(x)=\frac{x-5+5}{x-5}\) (C) \(g(x)=\frac{x}{x-5+5}\) (D) \(g(x)=\frac{x}{x}-\frac{x}{5}\)

THOUGHT PROVOKING The weight \(w\) (in pounds) of an object varies inversely as the square of the distance \(d\) (in miles) of the object from the center of Earth. At sea level ( 3978 miles from the center of the Earth), an astronaut weighs 210 pounds. How much does the astronaut weigh 200 miles above sea level?

Solve the equation by using the LCD. Check your solution(s). $$\frac{5}{x^2+x-6}=2+\frac{x-3}{x-2}$$

Find the quotient. $$ \frac{2 x^2-12 x}{x^2-7 x+6} \div \frac{2 x}{3 x-3} $$

In Exercises 25–32, graph the function. State the domain and range. $$ h(x)=\frac{-5 x}{-2 x-3} $$

OPEN-ENDED Describe a real-life situation that can be modeled by an inverse variation equation.

Find the quotient. $$ \frac{x^2-x-6}{x+4} \div\left(x^2-6 x+9\right) $$

In Exercises 25–32, graph the function. State the domain and range. $$ y=\frac{-2 x+3}{-x+10} $$

CRITICAL THINKING Suppose \(x\) varies inversely with \(y\) and \(y\) varies inversely with \(z\). How does \(x\) vary with \(z\) ? Justify your answer.

Find the quotient. $$ \frac{x^2-5 x-36}{x+2} \div\left(x^2-18 x+81\right) $$

Rewrite the function \(g\) in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). \(g(x)=\frac{12 x}{x-5}\)

You can paint a room in 8 hours. Working together, you and your friend can paint the room in just 5 hours. a. Let \(t\) be the time (in hours) your friend would take to paint the room when working alone. Copy and complete the table. \((\) Hint \(:(\) Work done \()=(\) Work rate \() \times(\) Time \())\) b. Explain what the sum of the expressions represents in the last column. Write and solve an equation to find how long your friend would take to paint the room when working alone.

Find the quotient. $$ \frac{x^2+9 x+18}{x^2+6 x+8} \div \frac{x^2-3 x-18}{x^2+2 x-8} $$

\(\left(x^2+2 x-99\right) \div(x+11)\)

Find the quotient. $$ \frac{x^2-3 x-40}{x^2+8 x-20} \div \frac{x^2+13 x+40}{x^2+12 x+20} $$

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{2 x-4}{x-5} $$

Rewrite the function \(g\) in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). \(g(x)=\frac{2 x+3}{x}\)

\(\left(3 x^4-13 x^2-x^3+6 x-30\right) \div\left(3 x^2-x+5\right)\) Graph the function. Then state the domain and range. (Section 6.4)

Give an example of a rational equation that you would solve using cross multiplication and one that you would solve using the LCD. Explain your reasoning.

Manufacturers often package products in a way that uses the least amount of material. One measure of the effi ciency of a package is the ratio of its surface area to its volume. The smaller the ratio, the more effi cient the packaging. a. Write an expression for the efficiency ratio \(\frac{S}{V}\) of a cylindrical package. b. Find the efficiency ratio for each cylindrical can listed in the table. $$ \begin{array}{|l|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{\underline{\phantom{xx}}} & \text { Soup } & \text { Coffee } & \text { Paint } \\ \hline \text { Height, } \boldsymbol{x} & 10.2 \mathrm{~cm} & 15.9 \mathrm{~cm} & 19.4 \mathrm{~cm} \\ \hline \text { Radius, } \boldsymbol{r} & 3.4 \mathrm{~cm} & 7.8 \mathrm{~cm} & 8.4 \mathrm{~cm} \\ \hline \end{array} $$ c. Rank the three cans in part (b) according to efficiency. Explain.

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{4 x-11}{x-2} $$

Rewrite the function \(g\) in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). \(g(x)=\frac{4 x-6}{x}\)

\(f(x)=5^x+4\)

Describe a real-life situation that can be modeled by a rational equation. Justify your answer.

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{x+18}{x-6} $$

Rewrite the function \(g\) in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). \(g(x)=\frac{3 x+11}{x-3}\)

\(g(x)=e^{x-1}\)

The total amount I (in millions of dollars) of healthcare expenditures and the residential population P (in millions) in the United States can be modeled by $$ \begin{aligned} &I=\frac{171,000 t+1,361,000}{1+0.018 t} \text { and } \\ &P=2.96 t+278.649 \end{aligned} $$ where \(t\) is the number of years since 2000. Find a model \(M\) for the annual healthcare expenditures per resident. Estimate the annual healthcare expenditures per resident in 2010.

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{2}{x-4}$$

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{x+2}{x-8} $$

\(y=\ln 3 x-6\)

The total amount I (in millions of dollars) of school expenditures from prekindergarten to a college level and the enrollment P (in millions) in prekindergarten through college in the United States can be modeled by $$ I=\frac{17,913 t+709,569}{1-0.028 t} \text { and } P=0.5906 t+70.219 $$ where \(t\) is the number of years since 2001. Find a model \(M\) for the annual education expenditures per student. Estimate the annual education expenditures per student in 2009. cant copy image

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{7}{x+6}$$

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{7 x-20}{x+13} $$