Chapter 2

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$\frac{5}{x}=1+\frac{3}{x+2}$$

A room is 1.5 times as long as it is wide, and its perimeter is 25 meters. (a) Draw a diagram that gives a visual representation of the problem. Identify the length as \(l\) and the width as \(w\) (b) Write \(l\) in terms of \(w\) and write an equation for the perimeter in terms of \(w\) (c) Find the dimensions of the room.

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$\sqrt{-29}$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$\frac{4}{x+1}-\frac{3}{x+2}=1$$

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$2 x^{2}-x-1=0$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$\frac{3}{x}+1=\frac{3}{x-1}$$

A picture frame has a total perimeter of 3 meters. The height of the frame is \(\frac{2}{3}\) times its width. (a) Draw a diagram that gives a visual representation of the problem. Identify the width as \(w\) and the height as \(h\) (b) Write \(h\) in terms of \(w\) and write an equation for the perimeter in terms of \(w\) (c) Find the dimensions of the picture frame.

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$\sqrt{-10}$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$\frac{1}{t^{2}}+\frac{8}{t}+15=0$$

Use absolute value notation to define the interval (or pair of intervals) on the real number line. All real numbers less than 10 units from 6

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$\frac{4}{7} x^{2}-8 x+28=0$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$-|x+1|=-3$$

To get an \(\mathrm{A}\) in a course, you must have an average of at least 90 on four tests of 100 points each. The scores on your first three tests were \(93,91,\) and 84 (a) Write a verbal model for the test average for the course. (b) What is the least you can score on the fourth test to get an \(A\) in the course?

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$9-\sqrt{6} i$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$6-\frac{1}{x}-\frac{1}{x^{2}}=0$$

Use absolute value notation to define the interval (or pair of intervals) on the real number line. All real numbers no more than 8 units from -5

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$\frac{1}{3} x^{2}-5 x+25=0$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$-|x-2|=-6$$

A store generates Monday through Thursday sales of \(\$ 150, \$ 125, \$ 75,\) and \(\$ 180 .\) What sales on Friday would give a weekday average of \(\$ 150 ?\)

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-8+\sqrt{15} i$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$\frac{x-2}{x}-\frac{1}{x+2}=0$$

Use absolute value notation to define the interval (or pair of intervals) on the real number line. All real numbers more than 3 units from -1

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$-0.2 x^{2}+1.2 x-8=0$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$|3 x-2|-1=4$$

A salesperson is driving from the office to a client, a distance of about 250 kilometers. After 30 minutes, the salesperson passes a town that is 50 kilometers from the office. Assuming the salesperson continues at the same constant speed, how long will it take to drive from the office to the client?

Write the quotient in standard form. $$\frac{6}{i}$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$\frac{x}{x^{2}-4}+\frac{1}{x+2}=3$$

Use absolute value notation to define the interval (or pair of intervals) on the real number line. All real numbers at least 5 units from 3

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$9+2.4 x-8.3 x^{2}=0$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$|4 x+1|+2=8$$

On the first part of a 336 -mile trip, a salesperson averaged 58 miles per hour. The salesperson averaged only 52 miles per hour on the last part of the trip because of an increased volume of traffic. The total time of the trip was 6 hours. Find the amount of time at each of the two speeds.

Write the quotient in standard form. $$-\frac{5}{2 i}$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$6\left(\frac{s}{s+1}\right)^{2}+5\left(\frac{s}{s+1}\right)-6=0$$

Determine the intervals on which the polynomial is entirely negative and those on which it is entirely positive. $$x^{2}-4 x-5$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$x^{2}-9 x+19=0$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$\sqrt{x-2}=3$$

A truck driver traveled at an average speed of 55 miles per hour on a 200 -mile trip to pick up a load of freight. On the return trip (with the truck fully loaded), the average speed was 40 miles per hour. Find the average speed for the round trip.

Write the quotient in standard form. $$\frac{2}{4-5 i}$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$8\left(\frac{t}{t-1}\right)^{2}-2\left(\frac{t}{t-1}\right)-3=0$$

Determine the intervals on which the polynomial is entirely negative and those on which it is entirely positive. $$x^{2}-3 x-4$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$x^{2}-10 x+22=0$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$\sqrt{x-4}=8$$

You are driving on a Canadian freeway to a town that is 300 kilometers from your home. After 30 minutes, you pass a freeway exit that you know is 50 kilometers from your home. Assuming that you continue at the same constant speed, how long will it take for the entire trip?

Write the quotient in standard form. $$\frac{3}{1-i}$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|2 x-5|=11$$

Determine the intervals on which the polynomial is entirely negative and those on which it is entirely positive. $$2 x^{2}-4 x-3$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$x^{2}+3 x=-8$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$2-\sqrt{x+5}=1$$

To determine the height of a pine tree, you measure the shadow cast by the tree and find it to be 20 feet long. Then you measure the shadow cast by a 36 -inch-tall oak sapling and find it to be 24 inches long (see figure). Estimate the height of the pine tree.

Write the quotient in standard form. $$ \frac{3-i}{3+i} $$