Chapter 8

Boyle’s Law states that when a sample of gas is kept at a constant temperature, the volume varies inversely with the pressure exerted on it. Write an equation for Boyle’s Law that expresses the variation in volume V as a function of pressure P.

Simplify each expression. $$ \frac{d-4}{d^{2}+2 d-8}-\frac{d+2}{d^{2}-16} $$

Simplify each expression. \(\frac{\frac{p^{3}}{2 q}}{-\frac{p^{2}}{4 q}}\)

In 2006, the cost to mail a first-class letter was 39¢ for any weight up to and including 1 ounce. Each additional ounce or part of an ounce added 24¢ to the cost. Make a graph showing the postal rates to mail any letter from 0 to 8 ounces.

Charles’ Law states that when a sample of gas is kept at a constant pressure, its volume V will increase directly as the temperature \(t\). Write an equation for Charles’ Law that expresses volume as a function

Simplify each expression. $$ \frac{\frac{1}{b+2}+\frac{1}{b-5}}{\frac{2 b^{2}-b-3}{b^{2}-3 b-10}} $$

Simplify each expression. \(\frac{\frac{m+n}{5}}{\frac{m^{2}+n^{2}}{5}}\)

CYCLING On a particular day, the wind added 3 kilometers per hour to Alfonso's rate when he was cycling with the wind and subtracted 3 kilometers per hour from his rate on his return trip. Alfonso found that in the same amount of time he could cycle 36 kilometers with the wind, he could go only 24 kilometers against the wind. What is his normal bicycling speed with no wind? Determine whether your answer is reasonable.

BASKETBALL For Exercises \(33-36,\) use the following information. Zonta plays basketball for Centerville High School. So far this season, she has made 6 out of 10 free throws. She is determined to improve her free-throw percentage. If she can make \(x\) consecutive free throws, her free-throw percentage can be determined using \(P(x)=\frac{6+x}{10+x}\) Graph the function.

A newspaper reported that the average American laughs 15 times per day. Write an equation to represent the average number of laughs produced by \(m\) household members during a period of \(d\) days.

Simplify each expression. $$ \frac{(x+y)\left(\frac{1}{x}-\frac{1}{y}\right)}{(x-y)\left(\frac{1}{x}+\frac{1}{y}\right)} $$

Simplify each expression. \(\frac{\frac{x+y}{2 x-y}}{\frac{x+y}{2 x+y}}\)

CHEMISTRY Kiara adds an 80\(\%\) acid solution to 5 milliters of solution that is 20\(\%\) acid. The function that represents the percent of acid in the resulting solution is \(f(x)=\frac{5(0.20)+x(0.80)}{5+x},\) where \(x\) is the amount of 80\(\%\) solution added. How much 80\(\%\) solution should be added to create a solution that is 50\(\%\) acid?

Find a counterexample to the statement All functions are continuous. Describe your function.

BASKETBALL For Exercises \(33-36,\) use the following information. Zonta plays basketball for Centerville High School. So far this season, she has made 6 out of 10 free throws. She is determined to improve her free-throw percentage. If she can make \(x\) consecutive free throws, her free-throw percentage can be determined using \(P(x)=\frac{6+x}{10+x}\) What part of the graph is meaningful in the context of the problem?

Simplify each expression. Under what conditions is \(\frac{x-4}{(x+5)(x-1)}\) undefined?

NUMBER THEORY The ratio of 3 more than a number to the square of 1 more than that number is less than \(1 .\) Find the numbers which satisfy this statement.

Identify each table of values as a type of function. A. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {7} \\ \hline-3 & {5} \\ \hline-1 & {3} \\ \hline 0 & {2} \\ \hline 1 & {3} \\ \hline 3 & {5} \\ \hline 5 & {7} \\ \hline 7 & {9} \\ \hline\end{array}\) B. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {24} \\ \hline-3 & {8} \\ \hline-1 & {0} \\ \hline 0 & {-1} \\ \hline 1 & {0} \\ \hline 3 & {8} \\ \hline 5 & {24} \\ \hline 7 & {48} \\ \hline\end{array}\) C. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-1.3 & {-1} \\ \hline-1.7 & {-1} \\ \hline 0 & {1} \\ \hline 0.8 & {1} \\ \hline 0.9 & {1} \\ \hline 0.9 & {1} \\ \hline 1 & {2} \\ \hline 1.5 & {2} \\ \hline 2.3 & {3} \\\ \hline\end{array}\) D. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {\text { undefined }} \\\ \hline-3 & {\text { undefined }} \\ \hline-1 & {\text { undefined }} \\\ \hline 0 & {0} \\ \hline 1 & {1} \\ \hline 4 & {2} \\ \hline 9 & {3} \\\ \hline 16 & {4} \\ \hline\end{array}\)

GEOMETRY Find the slope of a line that contains the points \(A\left(\frac{1}{p}, \frac{1}{q}\right)\) and \(B\left(\frac{1}{q}, \frac{1}{p}\right) .\) Write in simplest form.

Simplify each expression. For what values is \(\frac{2 d(d+1)}{(d+1)\left(d^{2}-4\right)}\) undefined?

STATISTICS For Exercises 36 and \(37,\) use the following information. A number \(x\) is the harmonic mean of \(y\) and \(z\) if \(\frac{1}{x}\) is the average of \(\frac{1}{y}\) and \(\frac{1}{z}\) Eight is the harmonic mean of 20 and what number?

Without graphing either function, explain how the graph of \(y=[x+2]-3\) is related to the graph of \(y=[x+1]-1\).

BASKETBALL For Exercises \(33-36,\) use the following information. Zonta plays basketball for Centerville High School. So far this season, she has made 6 out of 10 free throws. She is determined to improve her free-throw percentage. If she can make \(x\) consecutive free throws, her free-throw percentage can be determined using \(P(x)=\frac{6+x}{10+x}\) What is the equation of the horizontal asymptote? Explain its meaning with respect to Zonta's shooting percentage.

A parallelogram with an area of \(6 x^{2}-7 x-5\) square units has a base of \(3 x-5\) units. Determine the height of the parallelogram.

STATISTICS For Exercises 36 and \(37,\) use the following information. A number \(x\) is the harmonic mean of \(y\) and \(z\) if \(\frac{1}{x}\) is the average of \(\frac{1}{y}\) and \(\frac{1}{z}\) What is the harmonic mean of 5 and 8\(?\)

Find the LCM of each set of polynomials. $$ 9 p^{2} q^{3}, 6 p q^{4}, 4 p^{3} $$

Determine the equations of any vertical asymptotes and the values of \(x\) for any holes in the graph of each rational function. $$ f(x)=\frac{x^{2}-8 x+16}{x-4} $$

Find the LCM of each set of polynomials. $$ 2 t^{2}+t-3,2 t^{2}+5 t+3 $$

Determine the equations of any vertical asymptotes and the values of \(x\) for any holes in the graph of each rational function. $$ f(x)=\frac{x^{2}-3 x+2}{x-1} $$

Simplify each expression. \(\frac{\left(-3 x^{2} y\right)^{3}}{9 x^{2} y^{2}}\)

Find the LCM of each set of polynomials. $$ n^{2}-7 n+12, n^{2}-2 n-8 $$

Graph each rational function. $$ f(x)=\frac{3}{(x-1)(x+5)} $$

Paul drove from his house to work at an average speed of 40 miles per hour. The drive took him 15 minutes. If the drive home took him 20 minutes and he used the same route in reverse, what was his average speed going home?

Simplify each expression. \(\frac{\left(-2 r s^{2}\right)^{2}}{12 r^{2} s^{3}}\)

If \(x\) varies directly as \(y\) and \(y=\frac{1}{5}\) when \(x=11,\) find \(x\) when \(y=\frac{2}{5}\)

Simplify each expression. $$ \frac{5}{r}+7 $$

Graph each rational function. $$ f(x)=\frac{-1}{(x+2)(x-3)} $$

Many areas of Northern California depend on the snowpack of the Sierra Nevada Mountains for their water supply. If 250 cubic centimeters of snow will melt to 28 cubic centimeters of water, how much water does 900 cubic centimeters of snow produce?

Simplify each expression. \(\frac{\left(-5 m n^{2}\right)^{3}}{5 m^{2} n^{4}}\)

Graph each rational function. \(f(x)=\frac{3}{x+2}\)

Simplify each expression. $$ \frac{2 x}{3 y}+5 $$

Graph each rational function. $$ f(x)=\frac{x}{x^{2}-1} $$

Simplify each expression. \(\frac{y^{2}+4 y+4}{3 y^{2}+5 y-2}\)

ACT/SAT Amanda wanted to determine the average of her 6 test scores. She added the scores correctly to get \(T,\) but divided by 7 instead of 6 The result was 12 less than her actual average. Which equation could be used to determine the value of \(T ?\) $$ \begin{array}{l}{\text { A } 6 T+12=7 T} \\ {\text { B } \frac{T}{7}=\frac{T-12}{6}} \\ {\text { C } \frac{T}{7}+12=\frac{T}{6}} \\\ {\text { D } \frac{T}{6}=\frac{T-12}{7}}\end{array} $$

Graph each rational function. \(f(x)=\frac{8}{(x-1)(x+3)}\)

Simplify each expression. $$ \frac{3}{4 q}-\frac{2}{5 q}-\frac{1}{2 q} $$

Graph each rational function. $$ f(x)=\frac{x-1}{x^{2}-4} $$

Astronomers can use the brightness of two light sources, such as stars, to compare the distances from the light sources. The intensity, or brightness, of light I is inversely proportional to the square of the distance from the light source \(d .\) Write an equation that represents this situation.

Simplify each expression. \(\frac{a^{2}+2 a+1}{2 a^{2}+3 a+1}\)

REVIEW What is \(\frac{10 a^{-3}}{29 b^{4}} \div \frac{5 a^{-5}}{16 b^{-7}} ?\) $$ \begin{array}{l}{\mathbf{F} \frac{25 b^{3}}{232 a^{8}}} \\ {\text { G } \frac{25}{232 a^{2} b^{3}}} \\ {\text { H } \frac{32 b^{3}}{29 a^{8}}} \\\ {\text { J } \frac{32 a^{2}}{29 b^{11}}}\end{array} $$