Chapter 11

When you add rational expressions, you may need to factor a trinomial to find the LCD. Study the sample below. Then simplify the expressions in Exercises 46–49. $$\text { Sample: } \frac{2 x}{x^{2}-1}+\frac{3}{x^{2}+x-2}=\frac{2 x}{(x+1)(x-1)}+\frac{3}{(x-1)(x+2)}$$ The LCD is \((x+1)(x-1)(x+2)\) Note: If you just used \(\left(x^{2}-1\right)\left(x^{2}+x-2\right)\) as the common denominator, the factor \((x-1)\) would be included twice. $$\frac{2}{x-3}+\frac{x}{x^{2}+3 x-18}$$

Simplify. $$\left(-\frac{3}{4}\right)\left(\frac{3 y}{-5}\right)$$

Graph the function. Describe the domain. $$y=\frac{6}{x+9}-7$$

The charge for a cab ride is S11.50, and you give a 20\% tip. Using the model, find the total cost of the cab ride. Describe what the variable a represents. Model \(1: a\) is \(20 \%\) of \(\$ 11.50\)

Two events are independent if the probability that one event will occur is not affected by whether or not the other event occurs. For independent events \(\mathrm{A}\) and \(\mathrm{B}\), the probability that A and B will occur equals the probability of A times the probability of B. For example, if you draw a marble from the jar at the right, put it back, and then draw another one, the probability that both marbles are red is \(\frac{3}{5} \cdot \frac{3}{5}=\frac{9}{25}\). A bag contains \(n\) marbles. There are \(r\) blue marbles and the rest of the marbles are yellow. Find the probability of drawing a yellow marble followed by a blue marble if the first one is put back before drawing again.

Divide. Divide \(b^{2}-7 b-12\) by \(4 b+4\)

When you add rational expressions, you may need to factor a trinomial to find the LCD. Study the sample below. Then simplify the expressions in Exercises 46–49. $$\text { Sample: } \frac{2 x}{x^{2}-1}+\frac{3}{x^{2}+x-2}=\frac{2 x}{(x+1)(x-1)}+\frac{3}{(x-1)(x+2)}$$ The LCD is \((x+1)(x-1)(x+2)\) Note: If you just used \(\left(x^{2}-1\right)\left(x^{2}+x-2\right)\) as the common denominator, the factor \((x-1)\) would be included twice. $$\frac{2}{x^{2}-4}+\frac{3}{x^{2}+x-6}$$

Simplify. \(\frac{2 m}{3} \cdot 6 m^{2}\)

Graph the function. Describe the domain. $$y=\frac{3 x+11}{x+3}$$

Researchers studying fish populations at Dryden Lake in New York caught, marked, and then released 232 Chain pickerel. Later a sample of 329 Chain pickerel were caught and examined. Of these, 16 were found to be marked. Use the proportion below to estimate the total Chain pickerel population in the lake. \(\frac{\text { Marked pickerel in sample }}{\text { Total pickerel in sample }}=\frac{\text { Marked pickerel in lake }}{\text { Total pickerel in lake }}\)

The charge for a cab ride is S11.50, and you give a 20\% tip. Using the model, find the total cost of the cab ride. Describe what the variable a represents. Model \(2: a\) is \(120 \%\) of \(\$ 11.50 .\)

You are taking a trip on the highway in a car that gets a gas mileage of about 26 miles per gallon for highway driving. You start with a full tank of 12 gallons of gasoline. Find your rate of gas consumption (the gallons of gas used to drive 1 mile).

When you add rational expressions, you may need to factor a trinomial to find the LCD. Study the sample below. Then simplify the expressions in Exercises 46–49. $$\text { Sample: } \frac{2 x}{x^{2}-1}+\frac{3}{x^{2}+x-2}=\frac{2 x}{(x+1)(x-1)}+\frac{3}{(x-1)(x+2)}$$ The LCD is \((x+1)(x-1)(x+2)\) Note: If you just used \(\left(x^{2}-1\right)\left(x^{2}+x-2\right)\) as the common denominator, the factor \((x-1)\) would be included twice. $$\frac{7 x+2}{16-x^{2}}+\frac{7}{x-4}$$

Simplify. \(\frac{36}{45 a} \div \frac{-9 a}{5}\)

Graph the function. Describe the domain. $$y=\frac{-2 x+11}{x-5}$$

Find the least common denominator. $$\frac{3}{4}, \frac{2}{5}$$

When you add rational expressions, you may need to factor a trinomial to find the LCD. Study the sample below. Then simplify the expressions in Exercises 46–49. $$\text { Sample: } \frac{2 x}{x^{2}-1}+\frac{3}{x^{2}+x-2}=\frac{2 x}{(x+1)(x-1)}+\frac{3}{(x-1)(x+2)}$$ The LCD is \((x+1)(x-1)(x+2)\) Note: If you just used \(\left(x^{2}-1\right)\left(x^{2}+x-2\right)\) as the common denominator, the factor \((x-1)\) would be included twice. $$\frac{5 x-1}{2 x^{2}-7 x-15}-\frac{-3 x+4}{2 x^{2}+5 x+3}$$

Simplify. \(-18 c^{3} \div \frac{-27 c}{-4}\)

One way to prove that two proportions are equivalent is to apply the properties of equality to transform one of the proportions into the other proportion. Give a sequence of steps that transforms the proportion \(\frac{a}{b}=\frac{c}{d}\) into \(\frac{a}{c}=\frac{b}{d}\).

Graph the function. Describe the domain. $$y=\frac{9 x-6}{x-1}$$

Find the least common denominator. $$\frac{2}{9}, \frac{3}{18}$$

You are taking a trip on the highway in a car that gets a gas mileage of about 26 miles per gallon for highway driving. You start with a full tank of 12 gallons of gasoline. Do the variables g and m vary directly, inversely, or neither? Explain.

Find the LCD of \(\frac{-2}{x+9}\) and \(\frac{5 x}{x^{2}+9 x}\) (A) \(\frac{x-1}{(x-1)(2 x+1)}\) \((\mathbf{B})-\frac{x}{x-1}\) (c) \(\frac{2 x^{2}+1}{(x-1)(2 x+1)}\) (D) \(\frac{2 x^{2}-1}{(x-1)(2 x+1)}\)

Copy and complete the table. If necessary, round to the nearest tenth of a percent. $$\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline \text { Decimal } & ? & 0.2 & ? & 0.073 & 0.666 \ldots & ? & ? & 2 \\\\\hline \text { Percent } & 78 \% & ? & 3 \% & ? & ? & 176 \% & 110 \% & ? \\\\\hline\end{array}$$

Graph the function. Describe the domain. $$y=\frac{-5 x+19}{x-3}$$

Find the least common denominator. $$\frac{1}{16}, \frac{9}{20}$$

Use the models below that approximate spending in the United States from 1988 to \(1997 .\) Let \(t\) represent the number of years since 1988 . Dollars spent on exercise equipment (in millions): \(E=200 t+1400\) Total dollars spent on sports equipment (in millions): \(S=900 t+9900\) Write a rational model for the ratio of the money spent on exercise equipment to the total money spent on sports equipment. Simplify the model by dividing out the greatest common factor.

Use \(x\) to represent one dimension of a rectangle, and use \(y\) to represent the other dimension. a. Make a table of possible values of \(x\) and \(y\) if the area of the rectangle is 12 square inches. Then use your table to sketch a graph. b. Do \(x\) and \(y\) vary directly, inversely, or neither? Explain your reasoning. c. Make a table of possible values of \(x\) and \(y\) if the area of the rectangle is 24 square inches. Then use your table to sketch a graph in the same coordinate plane you used for your graph in part (a). d. CRITICAL THINKING How is the area of the first rectangle related to the area of the second rectangle? For a given value of \(x,\) how is the value of \(y\) for the first rectangle related to the value of \(y\) for the second rectangle? For a given value of \(y,\) how are the values of \(x\) related?

Simplify the expression \(\frac{x}{x-1}-\frac{1}{2 x+1}\) (A) \(\frac{x-1}{(x-1)(2 x+1)}\) (B) \(-\frac{x}{x-1}\) (C) \(\frac{2 x^{2}+1}{(x-1)(2 x+1)}\) (D) \(\frac{2 x^{2}-1}{(x-1)(2 x+1)}\)

Sketch the graph of the function. $$y=x^{2}$$

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$64$$

After 50 times at bat, a major league baseball player has a batting average of \(0.160 .\) How many consecutive hits must the player get to raise his batting average to \(0.250 ?\)

You earn \(10 \%\) more money at your summer job than your sister earns at her summer job. Does this mean that your sister earns \(10 \%\) less money than you? Explain your answer.

Find the least common denominator. $$\frac{14}{54}, \frac{31}{81}$$

Use the following information. In a direct variation, the ratio \(\frac{y}{x}\) is constant. If \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) are solutions of the equation \(\frac{y}{x}=k,\) then \(\frac{y_{1}}{x_{1}}=k\) and \(\frac{y_{2}}{x_{2}}=k .\) Use the proportion \(\frac{y_{1}}{x_{1}}=\frac{y_{2}}{x_{2}}\) to find the missing value. Find \(x_{2}\) when \(x_{1}=2, y_{1}=3,\) and \(y_{2}=6\)

You are making a 350 -mile car trip. You decide to drive a little faster to save time. Choose an expression for the time saved if the car's average speed \(s\) is increased by 5 miles per hour. $$\begin{array}{lllll} \text { (A) } \frac{350}{s+5} & \text { (B) } \frac{s+5}{350}-\frac{s}{350} & \text { (C) } \frac{350}{s}-\frac{350}{s+5} & \text { (D } 350(s+5)-350 s \end{array}$$

Sketch the graph of the function. $$y=4-x^{2}$$

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$-9$$

A student claims that if a price is now \(220 \%\) more than it was before, then it is \(320 \%\) of what it was before, and what it was before is \(31.25 \%\) of what it is now. Do you agree? Explain your answer.

Solve the equation. $$2 x^{2}+12 x-6=0$$

Use the following information. In a direct variation, the ratio \(\frac{y}{x}\) is constant. If \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) are solutions of the equation \(\frac{y}{x}=k,\) then \(\frac{y_{1}}{x_{1}}=k\) and \(\frac{y_{2}}{x_{2}}=k .\) Use the proportion \(\frac{y_{1}}{x_{1}}=\frac{y_{2}}{x_{2}}\) to find the missing value. Find \(y_{2}\) when \(x_{1}=-4, y_{1}=8,\) and \(x_{2}=-1\)

You will write and simplify a general expression for the average speed traveled when making a round trip. Let \(d\) represent the one-way distance. Let \(x\) represent the speed while traveling there and let \(y\) represent the speed while traveling back. Write an expression for the total time for the round trip. Use addition to write your answer as a single rational expression.

Sketch the graph of the function. $$y=\frac{1}{2} x^{2}$$

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$12$$

A library has received a single large contribution of \(\$ 5000 .\) A walkathon is also being held in which each sponsor will contribute \(\$ 10 .\) Let \(x\) represent the number of sponsors. Write a function that represents the average contribution per person, including the single large contributor. Sketch the graph of the function.

The variables \(x\) and \(y\) vary directly. Use the given values of the variables to write an equation that relates \(x\) and \(y .\) $$x=4, y=8$$

Solve the equation. $$x^{2}-6 x+7=0$$

Sketch the graph of the function. $$y=5 x^{2}+4 x-5$$

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$169$$

A library has received a single large contribution of \(\$ 5000 .\) A walkathon is also being held in which each sponsor will contribute \(\$ 10 .\) Let \(x\) represent the number of sponsors. Explain how such averages could be used to misrepresent actual contributions.