Chapter 2

Write each algebraic expression described. In \(2009,\) the number of graduate students at the University of Texas at Austin was approximately 28,000 fewer than the number of undergraduate students. If the number of undergraduate students was \(n\), how many graduate students attend UT Austin? (Source: University of Texas at Austin)

\(9(3 x+1)=4 x-5 x\)

The longest runway at Los Angeles International Airport has the shape of a rectangle and an area of 1,813,500 square feet. This runway is 150 feet wide. How long is the runway?

In Season 7 of American Idol, David Cook received 11.7 million more votes than runner-up David Archuleta. If 97.5 million votes were cast in the season finale, find the number of votes for each contestant. (Source: Los Angeles Times)

One more than five times a number is less than or equal to ten. Find all such numbers.

Solve. See Examples 1 through 7 $$ 3-\frac{1}{2} x=5 x-8 $$

The owner of a local chocolate shop wants to develop a new trail mix. How many pounds of chocolate-covered peanuts worth \(\$ 5\) a pound should be mixed with 10 pounds of granola bites worth \(\$ 2\) a pound to get a mixture worth \(\$ 3\) per pound?

Write each algebraic expression described. The longest interstate highway in the U.S. is I-90, which connects Seattle, Washington, and Boston, Massachusetts. The second longest interstate highway, I- 80 (connecting San Francisco, California, and Teaneck, New Jersey), is 178.5 miles shorter than I-90. If the length of I- 80 is \(m\) miles, express the length of I-90 as an algebraic expression in \(m\). (Source: U.S. Department of TransportationFederal Highway Administration)

\(7(2 x+1)=18 x-19 x\)

A geodesic dome, based on the design by Buckminster Fuller, is composed of two different types of triangular panels. One of these is an isosceles triangle. In one geodesic dome, the measure of the third angle is \(76.5^{\circ}\) more than the measure of either of the two equal angles. Find the measure of the three angles. (Source: Buckminster Fuller Institute)

Solve the following. For Exercises 61 and \(62,\) the solutions have been started for you. The perimeter of a rectangle is to be no greater than 100 centimeters and the width must be 15 centimeters. Find the maximum length of the rectangle.

Solve. See Examples 1 through 7 $$ \frac{3}{4} x-1+\frac{1}{2} x=\frac{5}{12} x+\frac{1}{6} $$

Write each algebraic expression described. The area of the Sahara Desert in Africa is 7 times the area of the Gobi Desert in Asia. If the area of the Gobi Desert is \(x\) square miles, express the area of the Sahara Desert as an algebraic expression in \(x\).

\(-\frac{3}{7} p=-2\)

The highest temperature ever recorded in Europe was \(122^{\circ} \mathrm{F}\) in Seville, Spain, in August of 1881 . Convert this record high temperature to Celsius.

The measures of the angles of a particular triangle are such that the second and third angles are each four times the measure of the smallest angle. Find the measures of the angles of this triangle.

Solve the following. For the solutions have been started for you. One side of a triangle is three times as long as another side, and the third side is 12 inches long. If the perimeter can be no longer than 32 inches, find the maximum lengths of the other two sides.

Solve. See Examples 1 through 7 $$ \frac{5}{9} x+2-\frac{1}{6} x=\frac{11}{18} x+\frac{1}{3} $$

Write each algebraic expression described. The largest meteorite in the world is the Hoba West located in Namibia. Its weight is 3 times the weight of the Armanty meteorite located in Outer Mongolia. If the weight of the Armanty meteorite is \(y\) kilograms, express the weight of the Hoba West meteorite as an algebraic expression in \(y\).

\(-\frac{4}{5} r=-5\)

The lowest temperature ever recorded in Oceania was \(-10^{\circ} \mathrm{C}\) at the Haleakala Summit in Maui, Hawaii, in January 1961. Convert this record low temperature to Fahrenheit.

Ben Holladay bowled 146 and 201 in his first two games. What must he bowl in his third game to have an average of at least \(180 ?\) (Hint: The average of a list of numbers is their sum divided by the number of numbers in the list.)

Solve. See Examples 1 through 7 $$ 3 x+\frac{5}{16}=\frac{3}{4}-\frac{1}{8} x-\frac{1}{2} $$

Place in the appropriate space to make each a true statement. $$ |-5| \quad-(-5) $$

Find each multiplicative inverse or reciprocal. $$ \frac{5}{8} $$

\(-\frac{4}{3} x=12\)

The CART FedEx Championship Series is an open-wheeled race car competition based in the United States. A CART car has a maximum length of 199 inches, a maximum width of 78.5 inches, and a maximum height of 33 inches. When the CART series travels to another country for a grand prix, teams must ship their cars. Find the volume of the smallest shipping crate needed to ship a CART car of maximum dimensions.

On an NBA team the two forwards measure \(6^{\prime} 8^{\prime \prime}\) and \(6^{\prime} 6^{\prime \prime}\) tall and the two guards measure \(6^{\prime} 0^{\prime \prime}\) and \(5^{\prime} 9^{\prime \prime}\) tall. How tall should the center be if they wish to have a starting team average height of at least \(6^{\prime} 5^{\prime \prime} ?\)

Solve. See Examples 1 through 7 $$ 2 x-\frac{1}{10}=\frac{2}{5}-\frac{1}{4} x-\frac{17}{20} $$

Find each multiplicative inverse or reciprocal. $$ \frac{7}{6} $$

\(-\frac{10}{3} x=30\)

On a road course, a CART car's speed can average up to around 105 mph. Based on this speed, how long would it take a CART driver to travel from Los Angeles to New York City, a distance of about 2810 miles by road, without stopping? Round to the nearest tenth of an hour.

Dennis and Nancy Wood are celebrating their 30 th wedding anniversary by having a reception at Tiffany Oaks reception hall. They have budgeted \(\$ 3000\) for their reception. If the reception hall charges a \(\$ 50.00\) cleanup fee plus \(\$ 34\) per person, find the greatest number of people that they may invite and still stay within their budget.

A plot of land is in the shape of a triangle. If one side is \(x\) meters, a second side is \((2 x-3)\) meters, and a third side is \((3 x-5)\) meters, express the perimeter of the lot as a simplified expression in \(x\).

Place in the appropriate space to make each a true statement. $$ (-3)^{2}-3^{2} $$

Find each multiplicative inverse or reciprocal. $$ 2 $$

\(-2 x-\frac{1}{2}=\frac{7}{2}\)

The Hoberman Sphere is a toy ball that expands and contracts. When it is completely closed, it has a diameter of 9.5 inches. Find the volume of the Hoberman Sphere when it is completely closed. Use 3.14 for \(\pi\). Round to the nearest whole cubic inch.

A surprise retirement party is being planned for Pratap Puri. A total of \(\$ 860\) has been collected for the event, which is to be held at a local reception hall. This reception hall charges a cleanup fee of \(\$ 40\) and \(\$ 15\) per person for drinks and light snacks. Find the greatest number of people that may be invited and still stay within the \(\$ 860\) budget.

A portion of a board has length \(x\) feet. The other part has length \((7 x-9)\) feet. Express the total length of the board as a simplified expression in \(x\).

Place in the appropriate space to make each a true statement. $$ |-2| \quad-|-2| $$

Find each multiplicative inverse or reciprocal. $$ 5 $$

\(-3 n-\frac{1}{3}=\frac{8}{3}\)

When the Hoberman Sphere (see Exercise 65 ) is completely expanded, its diameter is 30 inches. Find the volume of the Hoberman Sphere when it is completely expanded. Use 3.14 for \(\pi\).

A 150 -pound person uses 5.8 calories per minute when walking at a speed of 4 mph. How long must a person walk at this speed to use at least 200 calories? Round up to the nearest minute. (Source: Home \& Garden Bulletin No. 72)

Write each phrase as an algebraic expression. Use \(x\) for the unknown number. A number subtracted from -8

Is it possible to mix a \(10 \%\) acid solution and a \(40 \%\) acid solution to obtain a \(60 \%\) acid solution? Why or why not?

Find each multiplicative inverse or reciprocal. $$ -\frac{1}{9} $$

\(10=2 x-1\)

The average temperature on the planet Mercury is \(167^{\circ} \mathrm{C}\). Convert this temperature to degrees Fahrenheit. Round to the nearest degree.