Chapter 1

Solve the quadratic equation using any convenient method. \(26 x=8 x^{2}+15\)

Wholesale Price A store marks up a picture frame \(80 \%\) from its wholesale price. In a clearance sale, the price is discounted by \(40 \%\). The sale price is \(\$ 28.08\). What was the wholesale price of the picture frame?

A student states that the equation \(-3(x+2)=-3 x+6\) is an identity. Describe and correct the student's error.

Use a calculator to solve the inequality. (Round each number in your answer to two decimal places.) \(-0.5 x^{2}+12.5 x+1.6>0\)

Solve the inequality. Then graph the solution set on the real number line. \(|x+3|<5\)

An open box is to be made from a square piece of material by cutting two-inch squares from the corners and turning up the sides (see figure). The volume of the finished box is to be 200 cubic inches. Find the size of the original piece of material.

Solve the quadratic equation using any convenient method. \(50 x^{2}-60 x+10=0\)

Weekly Salary In Exercises 57 and 58 , use the following information to write a mathematical model and solve. Due to economic factors, your employer has reduced your weekly wage by \(15 \%\). Before the reduction, your weekly salary was \(\$ 425 .\) What is your reduced salary?

Explain why a solution of an equation involving fractional expressions may be extraneous.

Use a calculator to solve the inequality. (Round each number in your answer to two decimal places.) \(1.2 x^{2}+4.8 x+3.1<5.3\)

Solve the inequality. Then graph the solution set on the real number line. \(\left|\frac{2 x+1}{2}\right|<6\)

Solve the quadratic equation using any convenient method. \(9 x^{2}+12 x+3=0\)

Use the following information to write a mathematical model and solve. Due to economic factors, your employer has reduced your weekly wage by \(15 \%\). Before the reduction, your weekly salary was \(\$ 425 .\) What percent raise must you receive to bring your weekly salary back up to \(\$ 425\) ? Explain why the percent raise is different from the percent reduction.

Describe two methods you can use to check a solution of an equation involving fractional expressions.

Use a calculator to solve the inequality. (Round each number in your answer to two decimal places.) \(\frac{1}{2.3 x-5.2}>3.4\)

Solve the inequality. Then graph the solution set on the real number line. \(|x-20| \leq 4\)

Use a calculator to find the real solutions of the equation. (Round your answers to three decimal places.) \(3.2 x^{4}-1.5 x^{2}-2.1=0\)

You throw a coin straight up from the top of the Eiffel Tower in Paris with a velocity of 20 miles per hour. The building has a height of 984 feet. (a) Use the position equation to write a mathematical model for the height of the coin. (b) Find the height of the coin after 4 seconds. (c) How long will it take before the coin strikes the ground?

Solve the quadratic equation using any convenient method. \((x+3)^{2}-4=0\)

Travel Time You are driving to a college 150 miles from home. It takes 28 minutes to travel the first 30 miles. At this rate, how long is your entire trip?

What is meant by "equivalent equations"? Give an example of two equivalent equations.

Use a calculator to solve the inequality. (Round each number in your answer to two decimal places.) \(\frac{2}{3.1 x-3.7}>5.8\)

Solve the inequality. Then graph the solution set on the real number line. \(|x-7|<6\)

Use a calculator to find the real solutions of the equation. (Round your answers to three decimal places.) \(7.08 x^{6}+4.15 x^{3}-9.6=0\)

Solve the quadratic equation using any convenient method. \((x-2)^{2}-9=0\)

Travel Time Two friends fly from Denver to Orlando (a distance of 1526 miles). It takes 1 hour and 15 minutes to fly the first 500 miles. At this rate, how long is the entire flight?

For what value(s) of \(b\) does the equation \(7 x+3=7 x+b\) have infinitely many solutions? no solution?

A projectile is fired straight upward from ground level with an initial velocity of 200 feet per second. During what time period will its height exceed 400 feet?

Solve the inequality. Then graph the solution set on the real number line. \(|2 x-5|>6\)

Use a calculator to find the real solutions of the equation. (Round your answers to three decimal places.) \(1.8 x-6 \sqrt{x}-5.6=0\)

Solve the quadratic equation using any convenient method. \((x+1)^{2}=x^{2}\)

Travel Time Two cars start at the same time at a given point and travel in the same direction at constant speeds of 40 miles per hour and 55 miles per hour. After how long are the cars 5 miles apart?

In Exercises 61-66, use a calculator to solve the equation. (Round your solution to three decimal places.) \(0.275 x+0.725(500-x)=300\)

A projectile is fired straight upward from ground level with an initial velocity of 160 feet per second. During what time period will its height be less than 384 feet?

Solve the inequality. Then graph the solution set on the real number line. \(2|5-3 x|+7<21\)

Use a calculator to find the real solutions of the equation. (Round your answers to three decimal places.) \(4 x+8 \sqrt{x}+3.6=0\)

Two people are floating in a hot air balloon 200 feet above a lake. One person tosses out a coin with an initial velocity of 20 feet per second. One second later, the balloon is 20 feet higher and the other person drops another coin (see figure). The position equation for the first coin is \(s=-16 t^{2}+20 t+200\) and the position equation for the second coin is \(s=-16 t^{2}+220 .\) Which coin will hit the water first? (Hint: Remember that the first coin was tossed one second before the second coin was dropped.)

Solve the quadratic equation using any convenient method. \((x+1)^{2}=4 x^{2}\)

Catch-Up Time Students are traveling in two cars to a football game 135 miles away. One car travels at an average speed of 45 miles per hour. The second car starts \(\frac{1}{2}\) hour later and travels at an average speed of 55 miles per hour. How long will it take the second car to catch up to the first car?

Use a calculator to solve the equation. (Round your solution to three decimal places.) \(2.763-4.5(2.1 x-5.1432)=6.32 x+5\)

A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters (see figure). Within what bounds must the length be?

Solve the inequality. Then graph the solution set on the real number line. \(\left|\frac{x-3}{2}\right| \geq 5\)

Sharing the Cost A college charters a bus for $$\ 1700 to take a group of students to the Fiesta Bowl. When six more students join the trip, the cost per student decreases by $\$ 7.50 .$$ How many students were in the original group?

Consider the expression \((x+2)^{2}\). How would you convince someone in your class that \((x+2)^{2} \neq x^{2}+4 ?\) Give an argument based on the rules of algebra. Give an argument using your graphing utility.

Radio Waves Radio waves travel at the same speed as light, \(3.0 \times 10^{8}\) meters per second. Find the time required for a radio wave to travel from mission control in Houston to NASA astronauts on the surface of the moon \(3.84 \times 10^{8}\) meters away.

Use a calculator to solve the equation. (Round your solution to three decimal places.) \(\frac{x}{0.6321}+\frac{x}{0.0692}=1000\)

A rectangular room with a perimeter of 50 feet is to have an area of at least 120 square feet. Within what bounds must the length be?

Solve the inequality. Then graph the solution set on the real number line. \(\left|1-\frac{2 x}{3}\right|<1\)

Sharing the Cost Three students plan to share equally in the rent of an apartment. By adding a fourth person, each person could save $\$ 125 a month. How much is the monthly rent of the apartment?

Consider the expression \(\sqrt{a^{2}+b^{2}}\). How would you convince someone in your class that \(\sqrt{a^{2}+b^{2}} \neq a+b ?\) Give an argument based on the rules of algebra or geometry. Give an argument using your graphing utility.