Chapter 4

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{x y+x c+a y+a c}{x y-2 x c+a y-2 a c} \cdot \frac{2 x^{3}-8 x}{12 x^{3}+20 x^{2}-8 x}\)

For Problems 9-50, simplify each rational expression. \(\frac{6 x^{4}-11 x^{2}+4}{2 x^{4}+17 x^{2}-9}\)

An inlet pipe can fill a tank (see Figure 4.2) in \(10 \mathrm{~min}\) utes. A drain can empty the tank in 12 minutes. If the tank is empty, and both the pipe and drain are open, how long will it take before the tank overflows?

Set up an algebraic equation and solve each problem. One angle of a triangle has a measure of \(60^{\circ}\) and the measures of the other two angles are in the ratio of 2 to 3. Find the measures of the other two angles.

Perform the indicated divisions. $$ \left(x^{5}-1\right) \div(x-1) $$

Simplify each complex fraction. $$ \frac{\frac{3}{x}-\frac{2}{y}}{\frac{4}{y}-\frac{7}{x y}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{9 x y^{3}}-\frac{4}{3 x}+\frac{5}{2 y^{2}} $$

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{x^{2}-x}{4 y} \cdot \frac{10 x y^{2}}{2 x-2} \div \frac{3 x^{2}+3 x}{15 x^{2} y^{2}}\)

For Problems 9-50, simplify each rational expression. \(\frac{27 x^{4}-x}{6 x^{3}+10 x^{2}-4 x}\)

Barry can do a certain job in 3 hours, whereas it takes Sanchez 5 hours to do the same job. How long would it take them to do the job working together?

Set up an algebraic equation and solve each problem. The ratio of the complement of an angle to its supplement is 1 to 4 . Find the measure of the angle.

Simplify each complex fraction. $$ \frac{\frac{9}{x}+\frac{7}{x^{2}}}{\frac{5}{y}+\frac{3}{y^{2}}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{16 a^{2} b}+\frac{3 a}{20 b^{2}} $$

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{4 x y^{2}}{7 x} \cdot \frac{14 x^{3} y}{12 y} \div \frac{7 y}{9 x^{3}}\)

For Problems 9-50, simplify each rational expression. \(\frac{64 x^{4}+27 x}{12 x^{3}-27 x^{2}-27 x}\)

Connie can type 600 words in 5 minutes less than it takes Katie to type 600 words. If Connie types at a rate of 20 words per minute faster than Katie types, find the typing rate of each woman.

Set up an algebraic equation and solve each problem. The sum of a number and its reciprocal is \(\frac{53}{14}\). Find the number.

Perform the indicated divisions. $$ \left(x^{4}-1\right) \div(x+1) $$

Simplify each complex fraction. $$ \frac{\frac{6}{a}-\frac{5}{b^{2}}}{\frac{12}{a^{2}}+\frac{2}{b}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{2 x}{x-1}+\frac{3}{x} $$

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{a^{2}-4 a b+4 b^{2}}{6 a^{2}-4 a b} \cdot \frac{3 a^{2}+5 a b-2 b^{2}}{6 a^{2}+a b-b^{2}} \div \frac{a^{2}-4 b^{2}}{8 a+4 b}\)

For Problems 9-50, simplify each rational expression. \(\frac{-40 x^{3}+24 x^{2}+16 x}{20 x^{3}+28 x^{2}+8 x}\)

Walt can mow a lawn in 1 hour, and his son, Malik, can mow the same lawn in 50 minutes. One day Malik started mowing the lawn by himself and worked for 30 minutes. Then Walt joined him and they finished the lawn. How long did it take them to finish mowing the lawn after Walt started to help?

Set up an algebraic equation and solve each problem. The sum of two numbers is 80 . If the larger is divided by the smaller, the quotient is 7 , and the remainder is 8 . Find the numbers.

Perform the indicated divisions. $$ \left(x^{4}-1\right) \div(x-1) $$

Simplify each complex fraction. $$ \frac{\frac{4}{a b}-\frac{3}{b^{2}}}{\frac{1}{a}+\frac{3}{b}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3 x}{x-4}-\frac{2}{x} $$

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{2 x^{2}+3 x}{2 x^{3}-10 x^{2}} \cdot \frac{x^{2}-8 x+15}{3 x^{3}-27 x} \div \frac{14 x+21}{x^{2}-6 x-27}\)

For Problems 9-50, simplify each rational expression. \(\frac{-6 x^{3}-21 x^{2}+12 x}{-18 x^{3}-42 x^{2}+120 x}\)

Plane A can travel 1400 miles in 1 hour less time than it takes plane B to travel 2000 miles. The rate of plane B is 50 miles per hour greater than the rate of plane \(A\). Find the times and rates of both planes.

Set up an algebraic equation and solve each problem. If a home valued at \(\$ 150,000\) is assessed \(\$ 2500\) in real estate taxes, then how much, at the same rate, are the taxes on a home valued at \(\$ 210,000\) ?

Perform the indicated divisions. $$ \left(3 x^{4}+x^{3}-2 x^{2}-x+6\right) \div\left(x^{2}-1\right) $$

Simplify each complex fraction. $$ \frac{\frac{2}{x}-3}{\frac{3}{y}+4} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{a-2}{a}-\frac{3}{a+4} $$

Explain in your own words how to divide two rational expressions.

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{x y+a y+b x+a b}{x y+a y+c x+a c}\)

To travel 60 miles, it takes Sue, riding a moped, 2 hours less time than it takes Doreen to travel 50 miles riding a bicycle. Sue travels 10 miles per hour faster than Doreen. Find the times and rates of both girls.

Set up an algebraic equation and solve each problem. The ratio of male students to female students at a certain university is 5 to 7 . If there is a total of 16,200 students, find the number of male students and the number of female students.

Perform the indicated divisions. $$ \left(4 x^{3}-2 x^{2}+7 x-5\right) \div\left(x^{2}+2\right) $$

Simplify each complex fraction. $$ \frac{1+\frac{3}{x}}{1-\frac{6}{x}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{a+1}{a}-\frac{2}{a+1} $$

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{x y+2 y+3 x+6}{x y+2 y+4 x+8}\)

It takes Amy twice as long to deliver papers as it does Nancy. How long would it take each girl to deliver the papers by herself if they can deliver the papers together in 40 minutes?

Set up an algebraic equation and solve each problem. Suppose that, together, Laura and Tammy sold \(\$ 120.75\) worth of candy for the annual school fair. If the ratio of Tammy's sales to Laura's sales was 4 to 3 , how much did each sell?

For problems \(53-64\), use synthetic division to determine the quotient and remainder. $$ \left(x^{2}-8 x+12\right) \div(x-2) $$

Simplify each complex fraction. $$ \frac{3+\frac{2}{n+4}}{5-\frac{1}{n+4}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{-3}{4 n+5}-\frac{8}{3 n+5} $$

Give a step-by-step description of how to do the following multiplication problem. $$ \frac{x^{2}+5 x+6}{x^{2}-2 x-8} \cdot \frac{x^{2}-16}{16-x^{2}} $$

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{a x-3 x+2 a y-6 y}{2 a x-6 x+a y-3 y}\)

If two inlet pipes are both open, they can fill a pool in 1 hour and 12 minutes. One of the pipes can fill the pool by itself in 2 hours. How long would it take the other pipe to fill the pool by itself?