Chapter 9

Multiply or divide. State any restrictions on the variable. $$ \frac{x^{2}-x-2}{2 x^{2}-5 x+2} \div \frac{x^{2}-x-12}{2 x^{2}+5 x-3} $$

Solve each equation for the given variable. $$ \frac{q}{m}=\frac{2 V}{B^{2} r^{2}} ; B $$

Add or subtract. Simplify where possible. \(\frac{3}{4 x}-\frac{2}{x^{2}}\)

Business CDs can be manufactured for \(\$ .19\) each. The development cost is \(\$ 210,000\) . The first 500 discs are samples and will not be sold. a. Write a function for the average cost of a salable disc. Graph the function. b. What is the average cost if 5000 discs are produced? If \(15,000\) discs are produced? c. How many discs must be produced to bring the average cost under \(\$ 10 ?\) d. What are the vertical and horizontal asymptotes of the graph of the function?

Write each equation in the form \(y=\frac{k}{x}\). \(3 x y=12\)

Each ordered pair is from an inverse variation. Find the constant of variation. $$ \left(\frac{3}{8}, \frac{2}{3}\right) $$

Multiply or divide. State any restrictions on the variable. $$ \frac{2 x^{2}+5 x+2}{4 x^{2}-1} \cdot \frac{2 x^{2}+x-1}{x^{2}+x-2} $$

Anita and Fran have volunteered to contact every member of their organization by phone to inform them of an upcoming event. Fran can complete the calls in six days if she works alone. Anita can complete them in four days. How long will they take to complete the calls working together?

Add or subtract. Simplify where possible. \(\frac{3}{x+1}+\frac{x}{x-1}\)

Write each equation in the form \(y=\frac{k}{x}\). \(-7=5 x y\)

Each ordered pair is from an inverse variation. Find the constant of variation. $$ (\sqrt{2}, \sqrt{18}) $$

Simplify. State any restrictions on the variables. $$ \frac{\left(x^{2}-x\right)^{2}}{x(x-1)^{-2}\left(x^{2}+3 x-4\right)} $$

Add or subtract. Simplify where possible. \(\frac{2 x}{x^{2}-1}-\frac{1}{x^{2}}\)

Sketch the graph of each function. \(x y=3\)

Each ordered pair is from an inverse variation. Find the constant of variation. $$ (\sqrt{3}, \sqrt{27}) $$

Simplify. State any restrictions on the variables. $$ \frac{2 x+6}{(x-1)^{-1}\left(x^{2}+2 x-3\right)} $$

You are planning a school field trip to a local theater. It costs \(\$ 60\) to rent the bus. Each theater ticket costs \(55.50 .\) a. Write a function \(c(x)\) to represent the cost per student if \(x\) students sign up. b. How many students must sign up if the cost is to be no more than \(\quad \$ 10\) per student?

Add or subtract. Simplify where possible. \(\frac{4}{x^{2}-9}+\frac{7}{x+3}\)

Sketch the graph of each function. \(x y+5=0\)

Each ordered pair is from an inverse variation. Find the constant of variation. $$ (\sqrt{8}, \sqrt{32}) $$

Simplify. State any restrictions on the variables. $$ \frac{54 x^{3} y^{-1}}{3 x^{-2} y} $$

Woodworking A tapered cylinder is made by decreasing the radius of a rod continuously as you move from one end to the other. The rate at which it tapers is the taper per foot. You can calculate the taper per foot using the formula \(T=\frac{24(R-r)}{L} .\) The lengths \(R, r,\) and \(L\) are measured in inches. a. Solve this equation for \(L\) b. Find \(L\) if \(R=4\) in. \(r=3\) in.; and \(T=0.75,0.85,\) and 0.95

Add or subtract. Simplify where possible. \(\frac{x+2}{x-1}-\frac{x-3}{2 x+1}\)

Sketch the graph of each rational function. $$ y=\frac{2 x+3}{x-5} $$

Mechanics Gear A drives Gear B. Gear A has \(a\) teeth and speed \(r_{A}\) in revolutions per minute \((r p m) .\) Gear \(B\) has \(b\) teeth and speed \(r_{B} .\) The quantities are related by the formula \(a r_{\mathrm{A}}=b r_{\mathrm{B}}\) Gear \(\mathrm{A}\) has 60 teeth and speed 540 \(\mathrm{rpm} .\) Gear \(\mathrm{B}\) has 45 teeth. Find the speed of Gear \(\mathrm{B}\) .

A jar contains four blue marbles and two red marbles. Suppose you choose a marble at random, and do not replace it. Then you choose a second marble. Find the probability of each event. You select a blue marble and then a red marble.

Fuel Economy Suppose you drive an average of \(15,000\) miles per year, and your car gets 24 miles per gallon. Suppose gasoline costs \(\$ 1.60\) a gallon. a. How much money do you spend each year on gasoline? b. You plan to trade in your car for one that gets \(x\) more miles per gallon. Write an expression to represent the new vearly cost of gasoline. c. Write an expression to represent your savings on gasoline. d. Suppose you save \(\$ 200\) a year with the new car. How many miles per gallon does the new car get?

Add or subtract. Simplify where possible. \(\frac{x}{2 x^{2}-x}+\frac{1}{2 x}\)

Sketch the graph of each rational function. $$ y=\frac{x^{2}+6 x+9}{x+3} $$

Sketch the graph of each function. \(5 x y=2\)

Physics The force \(F\) of gravity on a rocket varies directly with its mass \(m\) and inversely with the square of its distance \(d\) from Earth. Write a model for this combined variation. \(k_{d^{2}}^{m}\)

A jar contains four blue marbles and two red marbles. Suppose you choose a marble at random, and do not replace it. Then you choose a second marble. Find the probability of each event. You select a red marble and then a blue marble.

Physics The acceleration of an object is a measure of how much its velocity changes in a given period of time. acceleration \(=\frac{\text { final velocity }-\text { initial velocity }}{\text { time }}\) Suppose you are riding a bicycle at 6 \(\mathrm{m} / \mathrm{s}\) . You step hard on the pedals and increase your speed to 12 \(\mathrm{m} / \mathrm{s}\) in about 5 \(\mathrm{s}\) . a. Find your acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) . b. A sedan can go from 0 to 60 \(\mathrm{mi} / \mathrm{h}\) in about 10 s. What is the acceleration in \(\mathrm{m} / \mathrm{s}^{2} ?(\text { Hints: } 1 \mathrm{mi} \approx 1609 \mathrm{m} ; 1 \mathrm{h}=3600 \mathrm{s} .)\)

Add or subtract. Simplify where possible. \(\frac{5 x}{x^{2}-x-6}-\frac{4}{x^{2}+4 x+4}\)

Sketch the graph of each rational function. $$ y=\frac{4 x^{2}-100}{2 x^{2}+x-15} $$

Sketch the graph of each function. \(10 x y=-4\)

Each pair of values is from a direct variation. Find the missing value. $$ (3,7),(8, y) $$

A jar contains four blue marbles and two red marbles. Suppose you choose a marble at random, and do not replace it. Then you choose a second marble. Find the probability of each event. One of the marbles you select is blue and the other is red.

a. Critical Thinking Simplify \(\frac{\left(2 x^{n}\right)^{2}-1}{2 x^{n}-1},\) where \(x\) is an integer and \(n\) is a positive integer. \((\text { Hint: Factor the numerator.) }\) b. Use the result from part (a) to show that the value of the given expression is always an odd integer.

Industry The average hourly wage \(H(x)\) of workers in an industry is modeled by the function \(H(x)=\frac{16.24 x}{0.062 x+39.42},\) where \(x\) represents the number of years since 1970 . a. In what year does the model predict that wages will be \(\$ 25 / \mathrm{h} ?\) b. Critical Thinking Is the prediction reasonable? Explain.

Add or subtract. Simplify where possible. \(3 x+\frac{x^{2}+5 x}{x^{2}-2}\)

Sketch the graph of each rational function. $$ y=-\frac{x}{(x-1)^{2}} $$

Each pair of values is from a direct variation. Find the missing value. $$ (2,5),(4, y) $$

Sketch the graph of each function. \(3 x y=-17\)

A jar contains four blue marbles and two red marbles. Suppose you choose a marble at random, and do not replace it. Then you choose a second marble. Find the probability of each event. Both of the marbles you select are red.

Use the fact that \(\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \div \frac{c}{d}\) to simplify each rational expression. State any restrictions on the variables. $$ \frac{\frac{8 x^{2} y}{x+1}}{\frac{6 x y^{2}}{x+1}} $$

Solve each equation. Check each solution. $$ \frac{15}{x}+\frac{9 x-7}{x+2}=9 $$

Add or subtract. Simplify where possible. \(4 y-\frac{y+2}{y^{2}+3 y}\)

Sketch the graph of each rational function. $$ y=\frac{2 x}{3 x-1} $$

Explain how knowing the asymptotes of a translation of \(y=\frac{k}{x}\) can help you graph the function. Include an example.