Chapter 6

Multiply, and then simplify, if possible. See Example 1. $$ \frac{3 a}{10} \cdot \frac{2}{15 a^{4}} $$

Let \(f(x)=\frac{2 x+1}{x^{2}+3 x-4}\). Find a. \(f(0)\) b. \(f(2)\) c. \(f(1)\)

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

Tell whether each relationship suggests direct or inverse variation. Lightning. The time it takes you to hear the lightning after a strike and your distance from the strike

Bottling. At a packaging plant, the older of two machines can fill \(5,000\) bottles of shampoo in 6 hours. A newer machine can fill \(5,000\) bottles in 4 hours. If both machines are used, how long will it take to fill \(5,000\) bottles of shampoo?

Complete each solution. $$\frac{x^{2}+3 x}{x-1}-\frac{2 x-1}{x-1}=\frac{x^{2}+3 x-(}{x-1}$$ $$=\frac{x^{2}+3 x-2 x\square 1}{x-1}$$ $$=\frac{x^{2}+\square+\square}{x-1}$$

Simplify. Write answers using positive exponents. \(\frac{25 x^{4} y^{7}}{5 x y^{9}}\)

Multiply, and then simplify, if possible. See Example 1. $$ \frac{4 p}{21} \cdot \frac{7}{12 p^{6}} $$

Fill in the blank: The rational expressions \(\frac{x+2}{x^{2}-4}\) and \(\frac{1}{x-2}\) are equivalent. They have the ___ those that make either denominator \(0 .\)

Use synthetic division to perform each division. See Example 1. $$ \left(3 x^{2}-13 x+12\right) \div(x-3) $$

Solve equation. \(\frac{4}{t+3}+\frac{8}{t^{2}-9}=\frac{2}{t-3}\)

Tell whether each relationship suggests direct or inverse variation. Desserts. The number of servings you can get from a wedding cake and the size of the piece that is served

Thrill Rides. At the end of an amusement park ride, a boat lands in a pool, splashing out a lot of water. Three inlet pipes, each working alone, can fill the pool in 10 seconds, 15 seconds, and 20 seconds, respectively. How long would it take to fill the pool if all three inlet pipes are used?

Simplify. Write answers using positive exponents. \(\frac{33 a^{2} b^{2}}{44 a^{4} b^{2}}\)

Multiply, and then simplify, if possible. See Example 1. $$ \frac{12 x^{61}}{7 y^{15}} \cdot \frac{y}{8 x^{27}} $$

Use synthetic division to perform each division. See Example 1. $$ \left(2 x^{2}-23 x+63\right) \div(x-7) $$

Solve equation. \(\frac{5}{x-1}=\frac{1}{x^{2}-1}+\frac{1}{x-1}\)

Fill in the blanks: $$\frac{-2 x^{2}-4 x}{(x+7)(x-9)}=\frac{\square(x+2)}{(x+7)(x-9)}=\square\frac{2 x(x+2)}{(x+7)(x-9)}$$

Tell whether each relationship suggests direct or inverse variation. Remodeling. The cost to remodel a house and the number of square feet to be added

Smoke Damage. Three ventilation fans, each working alone, can clear the smoke out of a room in 12 hours, 16 hours, and 24 hours, respectively. How long would it take to clear out the smoke in the room if all three fans are used?

Simplify. Write answers using positive exponents. \(\frac{63 a^{4}}{81 a^{6} b^{3}}\)

Multiply, and then simplify, if possible. See Example 1. $$ \frac{b^{65}}{27 a^{2}} \cdot \frac{18 a^{41}}{5 b^{90}} $$

What numbers are not included in each set of real numbers represented using interval notation? a. \((-\infty, 4) \cup(4, \infty)\) b. \((-\infty,-8) \cup(-8,0) \cup(0, \infty)\)

Solve equation. \(\frac{4}{x^{2}-4}-\frac{5}{x-2}=\frac{1}{x+2}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{8}{3 x}+\frac{5}{3 x}$$

Perform each division. \(\frac{4 x^{4}+6 x}{2}\)

Filling Ponds. One pipe can fill a pond in 3 weeks, and a second pipe can fill it in 5 weeks. However, evaporation and seepage can empty the pond in 10 weeks. If both pipes are used, how long will it take to fill the pond?

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{2}{x} $$

Solve equation. \(\frac{1}{m+3}-\frac{m}{m^{2}-9}=\frac{-2}{m-3}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{3}{4 y}+\frac{8}{4 y}$$

Perform each division. \(\frac{11 a^{3}-99 a^{2}}{11}\)

Housecleaning. Sally can clean the house in 6 hours, her father can clean the house in 4 hours, and her younger brother, Dennis, can completely mess up the house in 8 hours. If Sally and her father clean and Dennis plays, how long will it take to clean the house?

Multiply, and then simplify, if possible. See Example 2. $$ \frac{3 p^{2}}{6 p+24} \cdot \frac{p^{2}-16}{6 p} $$

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{8}{x-1} $$

Simplify each complex fraction. See Example 3. $$ \frac{4 p-\frac{4}{p}}{12-\frac{4}{p}} $$

Solve equation. \(\frac{2}{x-2}+\frac{10}{x+5}=\frac{2 x}{x^{2}+3 x-10}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{t}{4 r}+\frac{t}{4 r}$$

Solve each proportion. $$ \frac{x}{5}=\frac{15}{25} $$

Perform each division. \(\frac{4 x^{2}-x^{3}}{6 x}\)

Fine Dining. It takes a waiter 5 minutes less time than a busboy to fold the napkins used for the dinner seating in an upscale restaurant. Working together, they can fold the napkins in 6 minutes. How long would it take each person working alone to fold the napkins?

Multiply, and then simplify, if possible. See Example 2. $$ \frac{x^{2}+x-6}{5 x} \cdot \frac{5 x-10}{x+3} $$

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{2 x}{x+2} $$

Solve equation. \(\frac{2}{a+4}+\frac{2 a-1}{a^{2}+2 a-8}=\frac{1}{a-2}\)

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{16 x}{3 z^{2}}-\frac{x}{3 z^{2}}$$

Solve each proportion. $$ \frac{4}{y}=\frac{6}{27} $$

Perform each division. \(\frac{5 y^{4}+45 y^{3}}{15 y^{2}}\)

Fire Drill. If the east and west exit doors of a banquet hall are open, the occupants can clear out in 2 minutes. It takes 3 minutes longer to clear the hall if just the east door is open as it does if just the west door is open. How long does it take to clear the hall if just the west door is open?

Multiply, and then simplify, if possible. See Example 2. $$ \frac{z^{2}+4 z-5}{25 z-25} \cdot \frac{5 z}{z+5} $$

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{2 x+1}{x^{2}-2 x} $$

Simplify each complex fraction. See Example 4. $$ \frac{\frac{y}{x}-\frac{x}{y}}{\frac{1}{x}+\frac{1}{y}} $$