Chapter 6

Perform each division. \(\frac{x^{2}-5 x+6}{x-3}\)

Solve each proportion. $$ \frac{y}{4}=\frac{4}{y} $$

End Zones. One groundskeeper can paint the end zone of a football field in 2 hours. Another can paint it in 1 hour 20 minutes. How many minutes will it take them working together to paint the end zone?

Multiply, and then simplify, if possible. See Example 3. $$ \frac{9 x^{2}+3 x-20}{3 x^{2}-7 x+4} \cdot \frac{3 x^{2}-5 x+2}{9 x^{2}+18 x+5} $$

Simplify each rational expression. $$ \frac{12 x}{16 x^{7}} $$

Simplify each complex fraction. See Example 5. $$ \frac{x^{-2}-y^{-2}}{x^{-1}-y^{-1}} $$

Solve equation. \(\frac{2}{5 x-5}+\frac{x-2}{15}=\frac{4}{5 x-5}\)

Solve each proportion. $$ \frac{2}{c}=\frac{c-3}{2} $$

Perform each division. \(\frac{16 x^{2}-16 x-5}{4 x+1}\)

Truck Deliveries. \(\quad\) A trucker drove 75 miles to make a delivery at a mountain lodge and returned home on the same route. Because of foggy conditions, his average speed on the return trip was 10 mph less than his average speed going. If the return trip took 2 hours longer, how fast did he drive in each direction?

Multiply, and then simplify, if possible. See Example 3. $$ \frac{x^{2}+4 x y+4 y^{2}}{2 x^{2}+4 x y} \cdot \frac{3 x-6 y}{x^{2}-4 y^{2}} $$

Simplify each rational expression. $$ \frac{27 s t}{36 s t^{2}} $$

Solve equation. \(\frac{3}{2 x+4}=\frac{x-2}{2}+\frac{x-5}{2 x+4}\)

Solve each proportion. $$ \frac{2}{x+6}=\frac{-2 x}{5} $$

Perform each division. \(\frac{6 x^{2}-x-12}{2 x-3}\)

Moving Houses. A house mover towed a historic Victorian home 45 miles to locate it on a new site. On his return, without the heavy house in tow, his average speed was 30 mph faster and the trip was 2 hours shorter. How fast did he drive in each direction?

Simplify each rational expression. $$ \frac{49 x y^{2}}{21 x y} $$

Solve equation. \(\frac{1}{3 x-18}+\frac{5}{6-x}=\frac{1}{3}\)

Solve each proportion. $$ \frac{1}{x+3}=\frac{-2 x}{x+5} $$

Perform each division. \(\frac{6 x^{3}-x^{2}-6 x-9}{2 x-3}\)

Train Travel. An empty freight train traveled 60 miles from an auto assembly plant to an oil refinery. There, its tank cars were filled with petroleum products, and it returned on the same route to the plant. The total travel time for the train was \(5 \frac{1}{2}\) hours. If the train traveled 25 mph slower with the tank cars full, how fast did the train travel in each direction?

Multiply, and then simplify, if possible. See Example 3. $$ \frac{3 a^{2}+7 a b+2 b^{2}}{a^{2}+2 a b} \cdot \frac{a^{2}-a b}{3 a^{2}+a b} $$

Simplify each rational expression. $$ \frac{24 x^{3} y^{10}}{18 x^{4} y^{3}} $$

Use synthetic division to perform each division. See Example 3. Divide \(1-4 x+7 x^{2}+3 x^{3}\) by \(x+3\)

Solve equation. \(\frac{1}{2 x-16}+\frac{14}{8-x}=\frac{3}{2}\)

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

Perform each division. \(\frac{16 x^{3}+16 x^{2}-9 x-5}{4 x+5}\)

Boxing. For his morning workout, a boxer bicycles for 8 miles and then jogs back to camp along the same route. If he bicycles 6 mph faster than he jogs, and the entire workout lasts 2 hours, how fast does he jog?

Multiply, and then simplify, if possible. See Example 3. $$ \frac{a^{2}+3 a b+2 b^{2}}{a^{2}-3 a b-4 b^{2}} \cdot \frac{a^{2}-4 a b}{a b^{2}+2 b^{3}} $$

Simplify each rational expression. $$ \frac{15 a^{5} b^{4}}{21 a^{8} b^{3}} $$

Use synthetic division to perform each division. See Example 3. Divide \(27+x^{3}-17 x+8 x^{2}\) by \(x+10\)

Solve equation. \(\frac{7}{3 x-9}+\frac{1}{3-x}=\frac{4}{9}\)

Perform the operations and simplify the result when possible. See Example 3 . $$\frac{11}{5 m}-\frac{5}{6 m}$$

Solve each proportion. $$ \frac{2 b}{b+5}=\frac{-b}{3 b+8} $$

Perform each division. \(\frac{t^{3}+8 t^{2}+13 t+9}{t+6}\)

Rates of Speed. Two trains made the same 315 -mile run. since one train traveled 10 mph faster than the other, it arrived 2 hours earlier. Find the speed of each train.

Multiply, and then simplify, if possible. See Example 4. $$ 15 x\left(\frac{x+1}{15 x}\right) $$

Simplify each rational expression. $$ \frac{4 x^{2}}{2 x^{3}-12 x^{2}} $$

Simplify each complex fraction. See Example 6. $$ \frac{\frac{h}{h^{2}+3 h+2}}{\frac{4}{h+2}-\frac{4}{h+1}} $$

Solve equation. \(\frac{1}{2 d-4}-\frac{1}{2-d}=\frac{1}{4}\)

Solve each proportion. $$ \frac{-3 c}{c-2}=\frac{c}{c+2} $$

Perform each division. \(\frac{s^{3}+10 s^{2}+17 s+12}{s+8}\)

Deliveries. A FedEx delivery van traveled the 110 miles from Rockford to Chicago in 3 hours less time than it took a UPS van to travel the 275 miles from Rockford to St. Louis. If the vans traveled at the same average speed, for how long was the FedEx driver on the road?

Multiply, and then simplify, if possible. See Example 4. $$ 30 t\left(\frac{t-7}{30 t}\right) $$

Simplify each rational expression. $$ \frac{15 y^{2}}{5 y^{3}+15 y^{2}} $$

Simplify each complex fraction. See Example 6. $$ \frac{\frac{2}{x+3}-\frac{1}{x-3}}{\frac{3}{x^{2}-9}} $$

Solve each equation. If a solution is extraneous, so indicate. \(4-\frac{3 x}{x-9}=\frac{5 x-72}{x-9}\)

Express each verbal model in symbols. See Objectives 5 and 6. \(A\) varies directly as the square of \(p\)

Perform each division. \(\frac{6 x^{3}+11 x^{2}-19 x-2}{3 x-2}\)

Comparing Travel. A plane can fly 600 miles in the same time as it takes a car to go 240 miles. If the car travels 90 mph slower than the plane, find the speed of the plane.