Chapter 7

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{x+3}{(x+9)(x-2)}$$

Find the least common denominator of the rational expressions. $$\frac{5}{7(y+2)} \text { and } \frac{10}{y}$$

Simplify complex rational expression by the method of your choice. \(\frac{\frac{3}{4}-x}{\frac{3}{4}+x}\)

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

Multiply as indicated. $$\frac{x-3}{x+5} \cdot \frac{4 x+20}{9 x-27}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x}+\frac{2}{x}$$

Each exercise is a problem involving motion. The joys of the Pacific Coast! You drive 90 miles along the Pacific Coast Highway and then take a 5 -mile run along a hiking trail in Point Reyes National Seashore. Your driving rate is nine times that of your running rate. If the total time for driving and running is 3 hours, find the average rate driving and the average rate running.

Determine the constant of variation for each stated condition. \(T\) varies inversely as \(n,\) and \(T=4\) when \(n=24\)

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{4 x}{(3 x-17)(x+3)}$$

Find the least common denominator of the rational expressions. $$\frac{8}{11(y+5)} \text { and } \frac{12}{y}$$

Simplify complex rational expression by the method of your choice. \(\frac{\frac{2}{3}-x}{\frac{2}{3}+x}\)

Solve each rational equation. $$\frac{5}{x}+\frac{1}{3}=\frac{6}{x}$$

Multiply as indicated. $$\frac{x-2}{x+9} \cdot \frac{5 x+45}{2 x-4}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{5}{x}+\frac{13}{x}$$

Each exercise is a problem involving motion. The water's current is 2 miles per hour. A boat can travel 6 miles downstream, with the current, in the same amount of time it travels 4 miles upstream, against the current. What is the boat's average rate in still water?

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{4 x}{(3 x-17)(x+3)}$$

Find the least common denominator of the rational expressions. $$\frac{17}{x+4} \text { and } \frac{18}{x^{2}-16}$$

Simplify complex rational expression by the method of your choice. \(\frac{7-\frac{2}{x}}{5+\frac{1}{x}}\)

Solve each rational equation. $$\frac{2}{x}+3=\frac{5}{2 x}+\frac{13}{4}$$

Multiply as indicated. $$\frac{x^{2}+9 x+14}{x+7} \cdot \frac{1}{x+2}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{8}{9 x}+\frac{13}{9 x}$$

Each exercise is a problem involving motion. The water's current is 2 miles per hour. A canoe can travel 6 miles downstream, with the current, in the same amount of time it travels 2 miles upstream, against the current. What is the canoe's average rate in still water?

Use the four-step procedure for solving variation problems given on page 551 to solve Exercises \(9-12\). \(y\) varies directly as \(x . y=55\) when \(x=5 .\) Find \(y\) when \(x=13\)

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{8 x}{(4 x-19)(x+2)}$$

Find the least common denominator of the rational expressions. $$\frac{3}{x-6} \text { and } \frac{4}{x^{2}-36}$$

Simplify complex rational expression by the method of your choice. \(\frac{8+\frac{3}{x}}{1-\frac{7}{x}}\)

Solve each rational equation. $$\frac{7}{2 x}=\frac{5}{3 x}+\frac{22}{3}$$

Multiply as indicated. $$\frac{x^{2}+9 x+18}{x+6} \cdot \frac{1}{x+3}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{9 x}+\frac{11}{9 x}$$

Each exercise is a problem involving work. You must leave for campus in 10 minutes or you will be late for class. Unfortunately, you are snowed in. You can shovel the driveway in 20 minutes and your brother claims he can do it in 15 minutes. If you shovel together, how long will it take to clear the driveway? Will this give you enough time before you have to leave?

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{x+5}{x^{2}+x-12}$$

Simplify complex rational expression by the method of your choice. \(\frac{2+\frac{3}{y}}{1-\frac{7}{y}}\)

Find the least common denominator of the rational expressions. $$\frac{8}{y^{2}-9} \text { and } \frac{14}{y(y+3)}$$

Solve each rational equation. $$\frac{2}{3 x}+\frac{1}{4}=\frac{11}{6 x}-\frac{1}{3}$$

Multiply as indicated. $$\frac{x^{2}-25}{x^{2}-3 x-10} \cdot \frac{x+2}{x}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{5}{x+3}+\frac{4}{x+3}$$

Each exercise is a problem involving work. You promised your parents that you would wash the family car. You have not started the job and they are due home in 16 minutes. You can wash the car in 40 minutes and your sister claims she can do it in 30 minutes. If you work together, how long will it take to do the job? Will this give you enough time before your parents return?

Use the four-step procedure for solving variation problems given on page 551 to solve Exercises \(9-12\). \(y\) varies inversely as \(x . y=5\) when \(x=3 .\) Find \(y\) when \(x=9\)

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{7 x-14}{x^{2}-9 x+20}$$

Simplify complex rational expression by the method of your choice. \(\frac{4-\frac{7}{y}}{3-\frac{2}{y}}\)

Find the least common denominator of the rational expressions. $$\frac{14}{y^{2}-49} \text { and } \frac{12}{y(y-7)}$$

Solve each rational equation. $$\frac{5}{2 x}-\frac{8}{9}=\frac{1}{18}-\frac{1}{3 x}$$

Multiply as indicated. $$\frac{x^{2}-49}{x^{2}-4 x-21} \cdot \frac{x+3}{x}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{8}{x+6}+\frac{10}{x+6}$$

\- A person's fingernail growth, \(G\), in inches, varies directly ?s the number of weeks it has been growing, \(W\). Write an equation that expresses this relationship. Fingernails grow at a rate of about 0.02 inch per week. Substitute 0.02 for \(k,\) the constant of variation, in the equation in part (a) and write the equation for fingernail growth. =c. Substitute 52 for \(W\) to determine your fingernail growth at the end of one year if for some bizarre reason you decided not to cut them and they did not break.

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{x+5}{5}$$

Simplify complex rational expression by the method of your choice. \(\frac{\frac{1}{y}-\frac{3}{2}}{\frac{1}{y}+\frac{3}{4}}\)

Find the least common denominator of the rational expressions. $$\frac{7}{y^{2}-1} \text { and } \frac{y}{y^{2}-2 y+1}$$

Solve each rational equation. $$\frac{6}{x+3}=\frac{4}{x-3}$$

Multiply as indicated. $$\frac{4 y+30}{y^{2}-3 y} \cdot \frac{y-3}{2 y+15}$$