Chapter 11

Simplify the expression. $$\frac{x}{3 x^{2}+2 x-8} \cdot(3 x-4)$$

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=5, y=\frac{1}{3}$$

Simplify the expression. $$\frac{x^{2}+1}{x^{2}-4}+\frac{5 x}{x^{2}-4}-\frac{2 x+11}{x^{2}-4}$$

Solve the equation. $$\frac{x}{9}-\frac{8}{x}=\frac{1}{9}$$

Solve the proportion. Check for extraneous solutions. $$\frac{8}{x+2}=\frac{3}{x-1}$$

For what values of the variable is the rational expression undefined? $$\frac{x-3}{x^{2}+5 x-6}$$

Divide. Divide \(2 b^{2}-3 b-4\) by \(b-2\)

Simplify the expression. $$\frac{x+1}{x^{3}(3-x)} \div \frac{5}{x(x-3)}$$

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=30, y=7.5$$

Simplify the expression. $$\frac{x^{2}-9}{x+3}+\frac{x^{2}+9}{x-3}$$

Solve the equation. $$\frac{x+42}{x}=x$$

Solve the proportion. Check for extraneous solutions. $$\frac{x-3}{18}=\frac{3}{x}$$

Divide. Divide \(3 p^{2}+10 p+3\) by \(p+3\)

Solve the percent problem. 2 percent of what amount is \(\$ 200 ?\)

Simplify the expression. $$\left(4 x^{2}+x-3\right) \cdot \frac{1}{(4 x+3)(x-1)}$$

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=1.5, y=50$$

Simplify the expression. $$\frac{2}{x+1}+\frac{3}{x-2}+\frac{3}{x+4}$$

Solve the equation. $$\frac{2}{x}-\frac{x}{8}=\frac{3}{4}$$

Solve the proportion. Check for extraneous solutions. $$\frac{-2}{a-7}=\frac{a}{5}$$

Simplify the expression. $$\frac{x^{2}-8 x+15}{x^{2}-3 x} \div(3 x-15)$$

Divide. Divide \(5 g^{2}+14 g-2\) by \(g+3\)

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=45, y=\frac{3}{5}$$

Solve the equation. $$\frac{-3}{x+7}=\frac{2}{x+2}$$

Solve the proportion. Check for extraneous solutions. $$\frac{u}{3}=\frac{1}{2 u-1}$$

You are designing a game for a school carnival. Players will drop a coin into a basin of water, trying to hit a target on the bottom. The water is kept moving randomly, so the coin is equally likely to land anywhere. You use a rectangular basin twice as long as it is wide. You place the blue rectangular target an equal distance from each end. Express the two dimensions of the target in terms of the variables \(x\) and \(y\) THE GRAPH CANNOT COPY

Simplify the expression. $$\frac{6 x^{2}+7 x-33}{x+4} \div(6 x-11)$$

Divide. Divide \(c^{2}-25\) by \(c-5\)

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=10.5, y=7$$

Solve the equation. $$\frac{2}{x+3}+\frac{1}{x}=\frac{4}{3 x}$$

Solve the proportion. Check for extraneous solutions. $$\frac{d}{d+4}=\frac{d-2}{d}$$

Simplify the expression. $$\left(\frac{x^{2}}{5} \cdot \frac{x+2}{2}\right) \div \frac{x}{30}$$

Divide. Divide \(x^{2}-3 x-59\) by \(x-9\)

Make a table of values for x = 1, 2, 3, and 4. Use the table to sketch a graph. Decide whether x and y vary directly or inversely. $$y=\frac{4}{x}$$

In Exercises 34 and \(35,\) use the expression \(\frac{2 x-5}{x-2}\) and the table feature of a graphing calculator or spreadsheet software. Construct a table that shows the value of the numerator, the value of the denominator, and the value of the entire rational expression when the value of \(x\) is \(10,100,1000,10,000,100,000,\) and \(1,000,000\)

Solve the equation. $$\frac{10}{x+3}-\frac{3}{5}=\frac{10 x+1}{3 x+9}$$

Solve the proportion. Check for extraneous solutions. $$\frac{3 x}{4 x-1}=\frac{1}{x}$$

Simplify the expression. $$\left(\frac{2 x^{2}}{3} \cdot \frac{5}{x}\right) \div \frac{6 x^{2}}{25}$$

Divide. Divide \(d^{2}+15 d+45\) by \(d+5\)

Make a table of values for x = 1, 2, 3, and 4. Use the table to sketch a graph. Decide whether x and y vary directly or inversely. $$y=\frac{3}{2 x}$$

Use the expression \(\frac{2 x-5}{x-2}\) and the table feature of a graphing calculator or spreadsheet software. Use the table from Exercise \(34 .\) As \(x\) gets large, what happens to the values of the numerator? of the denominator? of the entire rational expression? Why do you think these results occur?

Solve the equation. $$\frac{x+3}{x-5}=\frac{56-3 x}{x^{2}-13 x+40}$$

Create three problems of the form \(\frac{a x^{2}+b x+c}{d x^{2}+e x+f}\) in which the numerator and the denominator have a common factor. Describe the process you used to create your problems.

Solve the proportion. Check for extraneous solutions. $$\frac{x-3}{x}=\frac{x}{x+6}$$

The models are based on data about train travel from 1990 to 1996 in the United States. Let \(t\) represent the number of years since \(1990 .\) D Source: Statistical Abstract of the United States Miles (in millions) traveled by passengers: \(\quad M=\frac{6300-800 t}{1-0.12 t}\) Passengers (in millions) who traveled by train: \(P=\frac{222-24 t}{10-t}\). Find a model for the average number of miles traveled per passenger.

Divide. Divide \(-x^{2}-6 x-16\) by \(x+2\)

Make a table of values for x = 1, 2, 3, and 4. Use the table to sketch a graph. Decide whether x and y vary directly or inversely. $$y=3 x$$

Use this information. A meteorite is equally likely to hit anywhere on Earth. The probability that a meteorite lands in the Torrid Zone is \(\frac{\text { Area of Torrid Zone }}{\text { Total surface area of Earth }}\) Let \(R\) represent Earth's radius. Write an expression to estimate the area of the Torrid Zone. You can think of the distance between the tropics (about 3250 miles ) as the height of a cylindrical belt around Earth at the equator. The length of the belt is Earth's circumference \(2 \pi R\) THE IMAGES CANNOT COPY

Solve the equation. $$\frac{8}{x+4}+1=\frac{5 x}{x^{2}-2 x-24}$$

Solve the proportion. Check for extraneous solutions. $$\frac{5}{m+1}=\frac{4 m}{m}$$

Divide. Divide \(-x^{2}+9 x-12\) by \(-x-2\)