Chapter 11

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

Solve. $$\frac{8}{r}+\frac{3}{r-1}=3$$

Multiply. $$\frac{4 a^{2} x-3 a^{2}}{2 b y+5 b} \cdot \frac{2 b^{3} y+5 b^{3}}{4 a x-3 a}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{x^{2}}{2 x-1}, \frac{x+1}{x+4}$$

Simplify. $$1-\frac{1}{1-\frac{1}{x}}$$

To assess the damage done by a fire, a forest ranger traveled 1080 mi by jet and then an additional 180 mi by helicopter. The rate of the jet was four times the rate of the helicopter. The entire trip took \(5 \mathrm{h}\). Find the rate of the jet.

Simplify. $$\frac{2 x+9}{9 x}-\frac{x-5}{5 x}$$

Solve. $$\frac{3}{5} y-\frac{1}{3}(1-y)=\frac{2 y-5}{15}$$

Two students are working with the equation \(A=P(1+i) .\) State whether the two students' answers are equivalent. a. When asked to solve the equation for \(i\), one student answered \(i=\frac{A}{P}-1\) and the other student answered \(i=\frac{A-P}{P}\) b. When asked to solve the equation for \(i\), one student answered \(i=-\frac{P-A}{P}\) and the other student answered \(i=\frac{A-P}{P}\)

Multiply. $$\frac{x^{2}+5 x+4}{x^{3} y^{2}} \cdot \frac{x^{2} y^{3}}{x^{2}+2 x+1}$$

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

Simplify. $$1-\frac{1}{1-\frac{1}{y+1}}$$

An engineer traveled 165 mi by car and then an additional 660 mi by plane. The rate of the plane was four times the rate of the car. The total trip took 6 h. Find the rate of the car.

Simplify. $$\frac{3 y-2}{12 y}-\frac{y-3}{18 y}$$

Solve. $$\frac{3}{4} a=\frac{1}{2}(3-a)+\frac{a-2}{4}$$

Multiply. $$\frac{x^{2}+x-2}{x y^{2}} \cdot \frac{x^{3} y}{x^{2}+5 x+6}$$

$$\text { Solve for } x: c x-y=b x+5$$

Write the fractions in terms of the LCM of the denominators. $$\frac{3 x}{x-5}, \frac{4}{x^{2}-25}$$

Simplify. $$\frac{a^{-1}-b^{-1}}{a^{-2}-b^{-2}}$$

After sailing \(15 \mathrm{mi}\), a sailor changed direction and increased the boat's speed by 2 mph. An additional 19 mi was sailed at the increased speed. The total sailing time was \(4 \mathrm{h}\). Find the rate of the boat for the first \(15 \mathrm{mi}\).

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

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

Multiply. $$\frac{x^{4} y^{2}}{x^{2}+3 x-28} \cdot \frac{x^{2}-49}{x y^{4}}$$

Solve the physics formula \(\frac{1}{R_{1}}+\frac{1}{R_{2}}=\frac{1}{R}\) for \(R_{2}\)

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

Simplify. $$\frac{x^{-2}-y^{-2}}{x^{-2} y^{-2}}$$

On a recent trip, a trucker traveled \(330 \mathrm{mi}\) at a constant rate. Because of road conditions, the trucker then reduced the speed by 25 mph. An additional 30 mi was traveled at the reduced rate. The entire trip took 7 h. Find the rate of the trucker for the first \(330 \mathrm{mi}\)

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

Solve. $$\frac{y+2}{y^{2}-y-2}+\frac{y+1}{y^{2}-4}=\frac{1}{y+1}$$

Multiply. $$\frac{x^{5} y^{3}}{x^{2}+13 x+30} \cdot \frac{x^{2}+2 x-3}{x^{7} y^{2}}$$

Business Break-even analysis is a method used to determine the sales volume required for a company to "break even," or experience neither a profit nor a loss on the sale of its product. The break-even point represents the number of units that must be made and sold for income from sales to equal the cost of producing the product. The break-even point can be calculated using the formula \(B=\frac{F}{S-V},\) where \(F\) is the fixed costs, \(S\) is the selling price per unit, and \(V\) is the variable costs per unit. a. Solve the formula \(B=\frac{F}{S-V}\) for \(S\) b. Use your answer to part (a) to find the selling price per button pinhole video spycam required for a company to break even. The fixed costs are \(\$ 15,000,\) the variable costs per spycam are \(\$ 60,\) and the company plans to make and sell 200 spycams. c. Use your answer to part (a) to find the selling price per spy camera video lighter required for a company to break even. The fixed costs are \(\$ 18,000,\) the variable costs per lighter are \(\$ 65,\) and the company plans to make and sell 600 lighters.

Write the fractions in terms of the LCM of the denominators. $$\frac{x-1}{x^{2}+2 x-15}, \frac{x}{x^{2}+6 x+5}$$

Commuting from work to home, a lab technician traveled \(10 \mathrm{mi}\) at a constant rate through congested traffic. Upon reaching the expressway, the technician increased the speed by 20 mph. An additional 20 mi was traveled at the increased speed. The total time for the trip was 1 h. At what rate did the technician travel through the congested traffic?

Simplify. $$\frac{x-10}{4 x^{2}}+\frac{x+1}{2 x}$$

Multiply. $$\frac{2 x^{2}-5 x}{2 x y+y} \cdot \frac{2 x y^{2}+y^{2}}{5 x^{2}-2 x^{3}}$$

Write each expression in terms of the LCM of the denominators. $$\frac{3}{10^{2}} ; \frac{5}{10^{4}}$$

As part of a conditioning program, a jogger ran \(8 \mathrm{mi}\) in the same amount of time it took a cyclist to ride \(20 \mathrm{mi}\). The rate of the cyclist was \(12 \mathrm{mph}\) faster than the rate of the jogger. Find the rate of the jogger and the rate of the cyclist.

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

Multiply. $$\frac{3 a^{3}+4 a^{2}}{5 a b-3 b} \cdot \frac{3 b^{3}-5 a b^{3}}{3 a^{2}+4 a}$$

Write each expression in terms of the LCM of the denominators. $$\frac{8}{10^{3}} ; \frac{9}{10^{5}}$$

In calm water, the rate of a small rental motorboat is 15 mph. The rate of the current on the river is 3 mph. How far down the river can a family travel and still return the boat in \(3 \mathrm{h} ?\)

Simplify. $$\frac{4}{x+4}-x$$

Multiply. $$\frac{x^{2}-2 x-24}{x^{2}-5 x-6} \cdot \frac{x^{2}+5 x+6}{x^{2}+6 x+8}$$

Write each expression in terms of the LCM of the denominators. $$b ; \frac{5}{b}$$

The rate of a small aircraft in calm air is 125 mph. If the wind is currently blowing south at a rate of \(15 \mathrm{mph}\), how far north can a pilot fly the plane and return it within \(2 \mathrm{h} ?\)

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

Multiply. $$\frac{x^{2}-8 x+7}{x^{2}+3 x-4} \cdot \frac{x^{2}+3 x-10}{x^{2}-9 x+14}$$

The speed of a boat in still water is \(20 \mathrm{mph}\). The Jacksons traveled \(75 \mathrm{mi}\) down the Woodset River in this boat in the same amount of time it took them to return \(45 \mathrm{mi}\) up the river. Find the rate of the river's current.

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

Multiply. $$\frac{x^{2}+2 x-35}{x^{2}+4 x-21} \cdot \frac{x^{2}+3 x-18}{x^{2}+9 x+18}$$