Chapter 6

Multiply, and then simplify, if possible. See Example 4. $$ 12 y\left(\frac{y+8}{6 y}\right) $$

Simplify each rational expression. $$ \frac{24 n^{4}}{16 n^{4}+24 n^{3}} $$

Solve each equation. If a solution is extraneous, so indicate. \(2-\frac{2 x}{x-10}=\frac{4 x-60}{x-10}\)

Comparing Travel. A bicyclist can travel 40 miles in the same time that a motorcyclist can travel 60 miles. If the bicyclist travels 12 mph slower than the motorcyclist, find the speed of the motorcyclist.

Simplify each rational expression. $$ \frac{18 m^{4}}{36 m^{4}-9 m^{3}} $$

Solve each equation. If a solution is extraneous, so indicate. \(\frac{6}{x+3}+\frac{48}{x^{2}-2 x-15}-\frac{7}{x-5}=0\)

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

Add or subtract, and then simplify, if possible. See Example 4. $$\frac{3}{x+2}+\frac{5}{x-4}$$

Express each verbal model in symbols. See Objectives 5 and 6. \(z\) varies inversely as the cube of \(t\)

Perform each division. \(\left(2 a+1+a^{2}\right) \div(a+1)\)

Boating. It takes 6 hours for a boater to travel 16 miles upstream and 16 miles back. If the speed of the boat in still water is \(6 \mathrm{mph},\) what is the speed of the current?

Simplify each rational expression. $$ \frac{2 x+18}{x^{2}-81} $$

Solve each equation. If a solution is extraneous, so indicate. \(\frac{3}{x-4}+\frac{2}{x+5}+\frac{18}{x^{2}+x-20}=0\)

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

Express each verbal model in symbols. See Objectives 5 and 6. \(v\) varies inversely as the square of \(r\)

Perform each division. \(\left(a-15+6 a^{2}\right) \div(2 a-3)\)

River Tours. A wave runner trip begins by going 60 miles upstream against a current. There, the driver turns around and returns with the current. If the still-water speed of the wave runner is set at 25 mph and the entire trip takes 5 hours, what is the speed of the current?

Simplify each rational expression. $$ \frac{6 x-12}{x^{2}-4} $$

Let \(P(x)=2 x^{3}-4 x^{2}+2 x-1 .\) Evaluate \(P(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ P(1) $$

Simplify each complex fraction. $$ \frac{\frac{x^{2}+5 x+6}{3 x y}}{\frac{9-x^{2}}{6 x y}} $$

Solve each formula for the specified variable. \(Q=\frac{A-I}{L}\) for \(A\) (from banking)

Express each verbal model in symbols. \(C\) varies jointly as \(x, y,\) and \(z\)

Perform each division. \(\left(6 y-4+10 y^{2}\right) \div(5 y-2)\)

Boating. A man can drive a motorboat 45 miles down the Colorado River in the same amount of time that he can drive 27 miles upstream. Find the speed of the current if the speed of the boat is 12 mph in still water.

Simplify each rational expression. $$ \frac{4 a^{2}-25}{20 a-50} $$

Simplify each complex fraction. $$ \frac{\frac{x-y}{x y}}{\frac{y-x}{x}} $$

Solve each formula for the specified variable. \(z=\frac{x-\bar{x}}{s}\) for \(x\) (from statistics)

Add or subtract, and then simplify, if possible. See Example 4. $$\frac{t}{t+2}+\frac{8}{t-2}$$

Express each verbal model in symbols. \(d\) varies jointly as \(r\) and \(t\)

Perform each division. \(\left(-10 x+x^{2}+16\right) \div(x-2)\)

Crop Dusting. \(\quad\) A helicopter spraying fertilizer over a field can fly 0.5 mile downwind in the same time as it can fly 0.4 mile upwind. Find the speed of the wind if the helicopter travels 45 mph in still air when dusting crops.

Multiply, and then simplify, if possible. See Example 4. $$ (2 a x-10 x+a-5) \cdot \frac{x}{2 x^{2}+x} $$

Simplify each rational expression. $$ \frac{9 b^{2}-16}{21 b+28} $$

Simplify each complex fraction. $$ \frac{1+\frac{6}{x}+\frac{8}{x^{2}}}{1+\frac{1}{x}-\frac{12}{x^{2}}} $$

Solve each formula for the specified variable. \(I=\frac{E}{R_{L}+r}\) for \(r\) (from physics)

Express each verbal model in symbols. \(P\) varies directly as the square of \(a\) and inversely as the cube of \(j\)

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

Catching a Plane. A walkway at an airport operates at a rate of 1.5 feet per second. Walking with the moving walkway, a man travels 65 feet in the same time that he could travel walking 35 feet in the opposite direction, against the walkway. What is the man's normal walking rate?

Simplify each rational expression. $$ \frac{5 x^{2}-10 x}{x^{2}-4 x+4} $$

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

Solve each formula for the specified variable. \(P=\frac{R-C}{n}\) for \(C\) (from business)

Perform each division. \(\frac{3+5 x+6 x^{3}+11 x^{2}}{3+2 x}\)

Express each verbal model in symbols. \(M\) varies inversely as the cube of \(n\) and jointly as \(x\) and the square of \(z\)

Kayaking. A kayaker can travel 1.2 miles downstream in the same time it takes him to go 0.4 miles upstream. If the river current flows at \(2 \mathrm{mph}\), what is the kayaker's speed in still water?

Simplify each rational expression. $$ \frac{x^{2}+6 x+9}{2 x^{2}+6 x} $$

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

Solve each formula for the specified variable. \(\mu_{R}=\frac{n_{1}\left(n_{1}+n_{2}+1\right)}{2}\) for \(n_{2}\) (from statistics)

Add or subtract, and then simplify, if possible. See Example 4. $$\frac{n+2}{n-4}-\frac{n+5}{n+4}$$

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ r=k t $$

In Example \(1,\) one crew could drywall a house in 4 days, and another crew could drywall the same house in 5 days. We were asked to find how long it would take them to drywall the house working together. Explain why each of the following approaches is incorrect. The time it would take to drywall the house: is the sum of the lengths of time it takes each crew to drywall the house: 4 days \(+5\) days \(=9\) days. drywall the house: 4 days \(+5\) days \(=9\) days. is the difference in lengths of time it takes each crew to drywall the house: 5 days \(-4\) days \(=1\) day. is the average of the lengths of time it takes each crew to drywall the house: \(\frac{4 \text { days }+5 \text { days }}{2}=\frac{9}{2}\) days \(=4 \frac{1}{2}\) days.