Chapter 11

Make a table of values for x = 1, 2, 3, and 4. Use the table to sketch a graph. Decide whether x and y vary directly or inversely. $$y=\frac{6}{x}$$

Solve the equation. $$\frac{x}{x-11}-1=\frac{22}{x^{2}-5 x-66}$$

Solve the proportion. Check for extraneous solutions. $$\frac{2}{3 t}=\frac{t-1}{t}$$

The models are based on data about train travel from 1990 to 1996 in the United States. Let \(t\) represent the number of years since \(1990 .\) D Source: Statistical Abstract of the United States Miles (in millions) traveled by passengers: \(\quad M=\frac{6300-800 t}{1-0.12 t}\) Passengers (in millions) who traveled by train: \(P=\frac{222-24 t}{10-t}\). Use the model to predict the average number of miles traveled per passenger in 2005.

Divide. Divide \(b^{2}-7 b+4\) by \(b+3\)

Decide if the data in the table show direct or inverse variation. Write an equation that relates the variables. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 3 & 5 & 10 & 0.5 \\\\\hline y & 5 & 15 & 25 & 50 & 2.5 \\\\\hline\end{array}$$

Solve the equation. $$\frac{2 x}{x+3}-\frac{x}{x+7}=\frac{x^{2}-1}{x^{2}+10 x+21}$$

Solve the proportion. Check for extraneous solutions. $$\frac{2}{6 x+1}=\frac{2 x}{1}$$

The models below are based on data collected by the Bureau of Economic Analysis from 1990 to 1997 in the United States. Let \(t\) represent the number of years since 1990 . Total sales (in billions of dollars) of services: \(S=\frac{1055+23 t}{1-0.04 t}\) Total sales (in billions of dollars) of hotel services: \(H=\frac{46+0.7 t}{1-0.04 t}\) Total sales (in billions of dollars) of auto repair services: \(A=\frac{48-t}{1-0.06 t}\) Find the total sales given by each model in \(1990 .\)

Divide. Divide \(5-7 m+3 m^{2}\) by \(m-3\)

Decide if the data in the table show direct or inverse variation. Write an equation that relates the variables. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 3 & 4 & 10 & 0.5 \\\\\hline y & 30 & 10 & 7.5 & 3 & 60 \\\\\hline\end{array}$$

Use the following information. You are choosing a business partner for a student lawn-care business you are starting. It takes you an average of 35 minutes to mow a lawn, so your rate is 1 lawn in 35 minutes or \(\frac{1}{35}\) of a lawn per minute. Let \(x\) represent the average time (in minutes) it takes a possible partner to mow a lawn. Write an expression for the partner's rate (that is, the part of a lawn the partner can mow in 1 minute). Then write an expression for the combined rate of you and your partner (the part of a lawn that you both can mow in 1 minute if you work together).

Solve the proportion. Check for extraneous solutions. $$\frac{-2}{q}=\frac{q+1}{q^{2}}$$

The models below are based on data collected by the Bureau of Economic Analysis from 1990 to 1997 in the United States. Let \(t\) represent the number of years since 1990 . Total sales (in billions of dollars) of services: \(S=\frac{1055+23 t}{1-0.04 t}\) Total sales (in billions of dollars) of hotel services: \(H=\frac{46+0.7 t}{1-0.04 t}\) Total sales (in billions of dollars) of auto repair services: \(A=\frac{48-t}{1-0.06 t}\) Find a model for the ratio of hotel service sales to total service industry sales. Was this ratio increasing or decreasing from 1990 to \(1997 ?\) Explain.

Divide. Divide \(x^{2}+9\) by \(-x-4\)

State whether the variables have direct variation, inverse variation, or neither. The area \(B\) of the base and the height \(h\) of a prism with a volume of 10 cubic units are related by the equation \(B h=10\).

Solve the proportion. Check for extraneous solutions. $$\frac{6}{19 n}=\frac{-2}{n^{2}+2}$$

The models below are based on data collected by the Bureau of Economic Analysis from 1990 to 1997 in the United States. Let \(t\) represent the number of years since 1990 . Total sales (in billions of dollars) of services: \(S=\frac{1055+23 \)\frac{48-2.92 t+0.04 t^{2}}{1055-40.3 t-1.38 t^{2}}\(t}{1-0.04 t}\) Total sales (in billions of dollars) of hotel services: \(H=\frac{46+0.7 t}{1-0.04 t}\) Total sales (in billions of dollars) of auto repair services: \(A=\frac{48-t}{1-0.06 t}\) Find a model for the ratio of auto service sales to total service industry sales. Was this ratio increasing or decreasing from 1990 to \(1997 ?\) Explain.

Divide. Divide \(4 x^{2}+12 x-10\) by \(x-2\)

State whether the variables have direct variation, inverse variation, or neither. The mass \(m\) and the volume \(V\) of a substance are related by the equation \(2 V=m,\) where 2 is the density of the substance.

Solve the proportion. $$\frac{x^{2}+5 x+6}{x^{2}-2 x-8}=\frac{x^{2}-4 x-5}{x^{2}-8 x+15} $$

Divide. Divide \(-5 m^{2}+2\) by \(m-1\)

State whether the variables have direct variation, inverse variation, or neither. Alicia cut a pizza into 8 pieces. The number of pieces \(d\) that Alicia ate for dinner and the number of pieces \(b\) that she can eat for breakfast are related by the equation \(b=8-d\)

Simplify the expression. $$\frac{2 x}{x+5}-\frac{3 x+2}{x+5}-\frac{4}{x+5}$$

Simplify. \left(-\frac{1}{2}\right)\left(\frac{2}{3}\right)

Graph the function. Describe the domain. $$y=\frac{1}{x}+4$$

Statement Explanation 1\. \(\frac{a c}{b c}=\frac{a}{b} \cdot \frac{?}{?}\) 1\. Apply the rule for multiplying rational expressions. 2\. \(\underline{?}=\frac{a}{b} \cdot \underline{?}\) 2\. Any nonzero number divided by itself is 1. 3\. \(\underline{?}=\frac{a}{b}\) 3\. Any nonzero number multiplied by 1 is itself. Copy and complete the proof to show why you can divide out common factors.

Divide. Divide \(4+11 q+6 q^{2}\) by \(2 q+1\)

State whether the variables have direct variation, inverse variation, or neither. The number of hours \(h\) that you must work to earn \(\$ 480\) and your hourly rate of pay \(p\) are related by the equation \(p h=480\)

Simplify the expression. $$\left(\frac{3 x^{2}}{56}\right)\left(\frac{3}{x}+\frac{5}{x}\right)$$

Simplify. (-15)\left(-\frac{5}{6}\right)

Graph the function. Describe the domain. $$y=\frac{1}{x-3}-8$$

A scale model uses a scale of \(\frac{1}{16}\) inch to represent 1 foot. Explain how you can use a proportion and cross products to show that a scale of \(\frac{1}{16}\) in. to 1 ft is the same as a scale of 1 in. to 192 in.

Statement Explanation 1\. \(\frac{a c}{b c}=\frac{a}{b} \cdot \frac{?}{?}\) 1\. Apply the rule for multiplying rational expressions. 2\. \(\underline{?}=\frac{a}{b} \cdot \underline{?}\) 2\. Any nonzero number divided by itself is 1. 3\. \(\underline{?}=\frac{a}{b}\) 3\. Any nonzero number multiplied by 1 is itself. Use the method from Exercise 42 to show that \(\frac{2 x-4}{x^{2}-4}=\frac{2}{x+2}\)

Divide. Divide \(4-s^{2}\) by \(s+5\)

Use the following information. When a person walks, the pressure P on each boot sole varies inversely with the area A of the sole. Denise is walking through deep snow, wearing boots that have a sole area of 29 square inches each. The boot-sole pressure is 4 pounds per square inch when she stands on one foot. The constant of variation is Denise’s weight in pounds. What is her weight?

Simplify the expression. $$\left(\frac{3 x-5}{x}+\frac{1}{x}\right) \div\left(\frac{x}{6 x-8}\right)$$

Simplify. \frac{2}{7} \div \frac{14}{24}

Graph the function. Describe the domain. $$y=\frac{2}{x-4}+6$$

Use the table. It shows the results of a survey in which 100 students were asked how they spent money last week. $$\begin{array}{|l|c|}\hline \quad \quad\quad\quad \text { Item } & \begin{array}{c}\text { Number } \\\\\text { of students }\end{array} \\\\\hline \text { Food } & 78 \\\\\hline \text { Clothes, accessories } & 20 \\\\\hline \text { Books, magazines, comics } & 15 \\\\\hline \text { Toys, stickers, games } & 14 \\ \hline \text { Movie tickets } & 14 \\\\\hline \text { Arcade games } & 14 \\\\\hline \text { Gifts } & 13 \\\\\hline \text { Movie rentals } & 13 \\\\\hline \text { Music } & 12 \\\\\hline \text { Footwear } & 11 \\\\\hline \text { Grooming products } & 11 \\\\\hline\end{array}$$ Estimate the number of students out of 500 that bought clothes or accessories in the last week.

In this survey the researchers tried to use a representative sample of people 18 years old and over in the United States. Would this sample be reasonable to use in predicting the responses of scientists? Explain.

Which of the following represents the expression \(\frac{x^{2}-3 x}{x^{2}-5 x+6} \cdot \frac{(x-2)^{2}}{2 x}\) in simplified form? (A) \(\frac{x(x-3)}{2}\) (B) \(\frac{x}{2}\) (C) \(\frac{x-2}{2}\) (D) \(\frac{x(x-3)}{x-2}\) (E) \(\frac{x^{2}-4 x+4}{x-2}\)

Use the following information. When a person walks, the pressure P on each boot sole varies inversely with the area A of the sole. Denise is walking through deep snow, wearing boots that have a sole area of 29 square inches each. The boot-sole pressure is 4 pounds per square inch when she stands on one foot. If Denise wears snowshoes, each with an area 11 times that of her boot soles, what is the snowshoe pressure when she stands on one foot?

Simplify the expression. $$\frac{x-2}{x+6} \div \frac{x+8}{4 x-24} \cdot \frac{x-8}{x-2}$$

Simplify. \frac{4}{9} \div(-36)

Graph the function. Describe the domain. $$y=-\frac{3}{x+1}+8$$

Use the table. It shows the results of a survey in which 100 students were asked how they spent money last week. $$\begin{array}{|l|c|}\hline \quad \quad\quad\quad \text { Item } & \begin{array}{c}\text { Number } \\\\\text { of students }\end{array} \\\\\hline \text { Food } & 78 \\\\\hline \text { Clothes, accessories } & 20 \\\\\hline \text { Books, magazines, comics } & 15 \\\\\hline \text { Toys, stickers, games } & 14 \\ \hline \text { Movie tickets } & 14 \\\\\hline \text { Arcade games } & 14 \\\\\hline \text { Gifts } & 13 \\\\\hline \text { Movie rentals } & 13 \\\\\hline \text { Music } & 12 \\\\\hline \text { Footwear } & 11 \\\\\hline \text { Grooming products } & 11 \\\\\hline\end{array}$$ Choose 3 items that were bought by different numbers of students. Based on the survey, how many students out of 20 would you predict to have bought each item?

You are shopping and find a coat that is on sale for \(30 \%\) off. It is regularly priced at \(\$ 80 .\) Your friend tells you that she saw the same coat for \(\$ 80\) in another store, but it was \(20 \%\) off plus an additional \(10 \%\) off. Will you save money by going to the other store? Explain why or why not.

Which product equals the quotient \((2 x+2) \div \frac{x^{2}+x}{4} ?\) (A) \(\frac{1}{2 x+2} \cdot \frac{x^{2}+x}{4}\) (B) \(\frac{2 x+2}{1} \cdot \frac{x^{2}+x}{4}\) (C) \(\frac{1}{2 x+2} \cdot \frac{4}{x^{2}+x}\) (D) \(\frac{2 x+2}{1} \cdot \frac{4}{x^{2}+x}\) (E) \(\frac{2 x+2}{2 x+2} \cdot \frac{4}{x^{2}+x}\)

Divide. Divide \(c^{2}-7 c+21\) by \(2 c-6\)