Chapter 11

For Exercises 21 to \(32,\) solve for \(y\). $$x+3 y=6$$

Simplify. $$\frac{x^{2}+3 x-28}{24-2 x-x^{2}}$$

Simplify. $$\frac{x-\frac{2}{2 x-3}}{2 x-1-\frac{8}{2 x-3}}$$

Two machines fill cereal boxes at the same rate. After the two machines work together for \(7 \mathrm{h}\), one machine breaks down. The second machine requires 14 more hours to finish filling the boxes. How long would it have taken one of the machines, working alone, to fill the boxes?

Simplify. $$\frac{7}{4 y}+\frac{11}{6 y}-\frac{8}{3 y}$$

Solve. $$2+\frac{3}{a-3}=\frac{a}{a-3}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{5 y}{6 x^{2}}, \frac{7}{9 x y}$$

For Exercises 21 to \(32,\) solve for \(y\). $$x+2 y=8$$

Simplify. $$\frac{x^{2}+7 x-8}{1+x-2 x^{2}}$$

Simplify. $$\frac{x+3-\frac{18}{2 x+1}}{x-\frac{6}{2 x+1}}$$

A mechanic requires \(2 \mathrm{h}\) to repair a transmission, whereas an apprentice requires \(6 \mathrm{h}\) to make the same repairs. The mechanic worked alone for \(1 \mathrm{h}\) and then stopped. How long will it take the apprentice, working alone, to complete the repairs?

Simplify. $$\frac{5}{3 x}-\frac{2}{x^{2}}+\frac{3}{2 x}$$

Solve. $$\frac{x}{x+4}=3-\frac{4}{x+4}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{y}{x(x-3)}, \frac{6}{x^{2}}$$

Multiply. $$\frac{8 x^{2}}{9 y^{3}} \cdot \frac{3 y^{2}}{4 x^{3}}$$

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

A large drain and a small drain are opened to drain a pool. The large drain can empty the pool in 6 h. After both drains have been open for 1 h, the large drain becomes clogged and is closed. The small drain remains open and requires 9 more hours to empty the pool. How long would it have taken the small drain, working alone, to empty the pool?

Simplify. $$\frac{6}{y^{2}}+\frac{3}{4 y}-\frac{2}{5 y}$$

Solve. $$\frac{x}{x-1}=\frac{8}{x+2}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{a}{y^{2}}, \frac{6}{y(y+5)}$$

Multiply. $$\frac{14 a^{2} b^{3}}{15 x^{5} y^{2}} \cdot \frac{25 x^{3} y}{16 a b}$$

Simplify. $$\frac{\frac{3}{n+1}+\frac{1}{n}}{\frac{2}{n+1}+\frac{3}{n}}$$

It takes Sam \(h\) hours to rake the yard, and it takes Emma \(k\) hours to rake the yard, where \(h>k .\) Let \(t\) be the amount of time it takes Sam and Emma to rake the yard together. Is \(t\) less than \(k\), between \(k\) and \(h\), or greater than \(k\) ?

Simplify. $$\frac{2}{x}-\frac{3}{2 y}+\frac{3}{5 x}-\frac{1}{4 y}$$

Solve. $$\frac{x}{x+12}=\frac{1}{x+5}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{9}{(x-1)^{2}}, \frac{6}{x(x-1)}$$

Multiply. $$\frac{12 x^{3} y^{4}}{7 a^{2} b^{3}} \cdot \frac{14 a^{3} b^{4}}{9 x^{2} y^{2}}$$

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

Simplify. $$\frac{5}{2 a}+\frac{7}{3 b}-\frac{2}{b}-\frac{3}{4 a}$$

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

Write the fractions in terms of the LCM of the denominators. $$\frac{a^{2}}{y(y+7)}, \frac{a}{(y+7)^{2}}$$

Multiply. $$\frac{18 a^{4} b^{2}}{25 x^{2} y^{3}} \cdot \frac{50 x^{5} y^{6}}{27 a^{6} b^{2}}$$

Simplify. $$\frac{\frac{4}{3 x+1}+\frac{3}{x}}{\frac{6}{x}-\frac{2}{3 x+1}}$$

Simplify. $$\frac{2 x+1}{3 x}+\frac{x-1}{5 x}$$

Solve. $$\frac{5}{3 n-8}=\frac{n}{n+2}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{3}{x-3}, \frac{5}{x(3-x)}$$

Multiply. $$\frac{3 x-6}{5 x-20} \cdot \frac{10 x-40}{27 x-54}$$

True or false? If the denominator of a complex fraction is the reciprocal of the numerator, then the complex fraction is equal to the square of its numerator.

Simplify. $$\frac{4 x-3}{6 x}+\frac{2 x+3}{4 x}$$

Solve. $$\frac{x}{x+4}=\frac{11}{x^{2}-16}+2$$

Write the fractions in terms of the LCM of the denominators. $$\frac{b}{y(y-4)}, \frac{b^{2}}{4-y}$$

Multiply. $$\frac{8 x-12}{14 x+7} \cdot \frac{42 x+21}{32 x-48}$$

Simplify. $$1+\frac{1}{1+\frac{1}{2}}$$

A camper drove 80 mi to a recreational area and then hiked 4 mi into the woods. The rate of the camper while driving was ten times the rate while hiking. The total time spent hiking and driving was 3 h. Find the rate at which the camper hiked.

Simplify. $$\frac{x-3}{6 x}+\frac{x+4}{8 x}$$

Solve. $$x-\frac{6}{x-3}=\frac{2 x}{x-3}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{x-2}{x+3}, \frac{x}{x-4}$$

Multiply. $$\frac{3 x^{2}+2 x}{2 x y-3 y} \cdot \frac{2 x y^{3}-3 y^{3}}{3 x^{3}+2 x^{2}}$$

Simplify. $$1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}$$

The president of a company traveled 1800 mi by jet and 300 \(\mathrm{mi}\) on a prop plane. The rate of the jet was four times the rate of the prop plane. The entire trip took 5 h. Find the rate of the jet.