1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 1 Equations and Inequalities

OBJECTIVES Linear Inequalities Learn the vocabulary for discussing inequalities. Solve and graph linear inequalities. Solve and graph a compound inequality. Solve and graph an inequality involving the reciprocal of a linear expression. SECTION © 2010 Pearson Education, Inc. All rights reserved

INEQUALITIES Equation Replace = by Inequality x = 5<x < 5 3x + 2 = 14≤3x + 2 ≤ 14 5x + 7 = 3x + 23>5x + 7 > 3x + 23 x 2 = 0≥x 2 ≥ 0 © 2010 Pearson Education, Inc. All rights reserved 3

Definitions An inequality is a statement that one algebraic expression is less than, or is less than or equal to, another algebraic expression. The domain of a variable in an inequality is the set of all real numbers for which both sides of the inequality are defined. The solutions of the inequality are the real numbers that result in a true statement when those numbers are substituted for the variable in the inequality. © 2010 Pearson Education, Inc. All rights reserved 4

Definitions To solve an inequality means to find all solutions of the inequality–that is, the solution set. The graph of the inequality x < 5 is the interval (–∞, 5) and is shown here. The solution sets are intervals, and we frequently graph the solutions sets for inequalities in one variable on a number line. x < 5, or (–∞, 5) ) 5 © 2010 Pearson Education, Inc. All rights reserved 5

Definitions A conditional inequality such as x < 5 has in its domain at least one solution and at least one number that is not a solution. An inconsistent inequality is one in which no real number satisfies it. An identity is an inequality that is satisfied by every real number in the domain. © 2010 Pearson Education, Inc. All rights reserved 6

NONNEGATIVE IDENTITY for any real number x. Because x 2 = x · x is the product of either (1) two positive factors, (2) two negative factors, or (3) two zero factors, x 2 is always either a positive number or zero. That is, x 2 is never negative, or is nonneagtive. © 2010 Pearson Education, Inc. All rights reserved 7

EQUIVALENT INEQUALITIES 1.Simplifying one or both sides of an inequality by combining like terms and eliminating parentheses 2.Adding or subtracting the same expression on both sides of the inequality Two inequalities that have exactly the same solution set are called equivalent inequalities. The following operations produce equivalent inequalities. © 2010 Pearson Education, Inc. All rights reserved 8

Sign of C InequalitySenseExample A < BA < B<3x < 12 positiveA · C < B · CUnchanged positiveUnchanged negativeA · C > B · CReversed negativeReversed If C represents a real number, then the following inequalities are all equivalent. © 2010 Pearson Education, Inc. All rights reserved 9

Linear Inequalities A linear inequality in one variable is an inequality that is equivalent to one of the forms ax + b < 0 or ax + b ≤ 0, where a and b represent real numbers and a ≠ 0. © 2010 Pearson Education, Inc. All rights reserved 10

EXAMPLE 1 Solving and Graphing Linear Inequalities Solve each inequality and graph its solution set. The solution set is {x|x < 1}, or (–∞, 1). Solution ) 01–1 © 2010 Pearson Education, Inc. All rights reserved 11

EXAMPLE 1 Solving and Graphing Linear Inequalities The solution set is {x|x ≥ 2}, or [2, ∞). Solution continued [ 120 © 2010 Pearson Education, Inc. All rights reserved 12

EXAMPLE 2 Calculating the Results of the Bermuda Triangle Experiment In the introduction to this section, we discussed an experiment to test the reliability of compass settings and flight by automatic pilot along one edge of the Bermuda Triangle. The plane is 150 miles along its path from Miami to Bermuda, cruising at 300 miles per hour, when it notifies the tower that it is now set on automatic pilot. The entire trip is 1035 miles, and we want to determine how much time we should let pass before we become concerned that the plane has encountered trouble. © 2010 Pearson Education, Inc. All rights reserved 13

EXAMPLE 2 Let t = time elapsed since plane on autopilot 300t = distance plane flown in t hours t = plane’s distance from Miami after t hours Solution Calculating the Results of the Bermuda Triangle Experiment Plane’s distance from Miami Distance from Miami to Bermuda ≥ © 2010 Pearson Education, Inc. All rights reserved 14

EXAMPLE 2 Solution continued Calculating the Results of the Bermuda Triangle Experiment Since 2.95 is roughly 3 hours, the tower will suspect trouble if the plane has not arrived in 3 hours. © 2010 Pearson Education, Inc. All rights reserved 15

EXAMPLE 3 Solving an Inequality Write the solution set of each inequality. Solution The last inequality is always true. So the solution set of the original inequality is (–∞, ∞). © 2010 Pearson Education, Inc. All rights reserved 16

EXAMPLE 3 Solving an Inequality Solution continued The resulting inequality is always false. So the solution set of the original inequality is . © 2010 Pearson Education, Inc. All rights reserved 17

EXAMPLE 4 Solving & Graphing a Compound Inequality Graph and write the solution set of the compound inequality. Solution © 2010 Pearson Education, Inc. All rights reserved 18

EXAMPLE 4 Solving & Graphing a Compound Inequality Solution continued Graph each inequality and select the union of the two intervals. x  –3 x > 2 x  –3 or x > 2 The solution set of the compound inequality is (–∞, –3]  (2, ∞). © 2010 Pearson Education, Inc. All rights reserved 19

EXAMPLE 5 Solving & Graphing a Compound Inequality Graph and write the solution set of the compound inequality. Solution © 2010 Pearson Education, Inc. All rights reserved 20

EXAMPLE 5 Solving & Graphing a Compound Inequality Solution continued Graph each inequality and select the interval common to both inequalities. x < 5 x ≥ –2 x < 5 and x ≥ –2 The solution set of the compound inequality is the interval [–2, 5). © 2010 Pearson Education, Inc. All rights reserved 21

EXAMPLE 6 Solving & Graphing a Compound Inequality Solve the inequality graph its solution set. and Solution First, solve the inequalities separately. © 2010 Pearson Education, Inc. All rights reserved 22

EXAMPLE 6 Solving & Graphing a Compound Inequality We can write this as {x| – 4 < x ≤ 3}. In interval notation we write (– 4, 3]. Solution continued The solution to the original inequalities consists of all real numbers x such that – 4 < x and x ≤ 3. ] 31 ( –104 – 4– 4 – 5– 52–2–3 © 2010 Pearson Education, Inc. All rights reserved 23

EXAMPLE 6 Solving & Graphing a Compound Inequality Solution continued We can accomplish this solution more efficiently by working on both inequalities at the same time as follows. © 2010 Pearson Education, Inc. All rights reserved 24

RECIPROCAL SIGN PROPERTY If x ≠ 0, x andare either both positive or negative. In symbols, if x > 0, then and if x < 0, then © 2010 Pearson Education, Inc. All rights reserved 25

EXAMPLE 7 Solving and Graphing an Inequality by Using the Reciprocal Sign Property Solve and graph Solution The solution set is {x| x >4}, or in interval notation (4, ∞). ( © 2010 Pearson Education, Inc. All rights reserved 26

EXAMPLE 8 Finding the Interval of Values for a Linear Expression If –2 < x < 5, find real numbers a and b so that a < 3x – 1 < b. Solution We have a = –7 and b = 14. © 2010 Pearson Education, Inc. All rights reserved 27

EXAMPLE 9 Finding a Fahrenheit Temperature from a Celsius Range The weather in London is predicted to range between 10º and 20º Celsius during the three- week period you will be working there. To decide what kind of clothes to bring, you want to convert the temperature range to Fahrenheit temperatures. The formula for converting Celsius temperature C to Fahrenheit temperature F is temperatures might you find in London during your stay there? What range of Fahrenheit © 2010 Pearson Education, Inc. All rights reserved 28

Let C = temperature in Celsius degrees. Solution For the three weeks under consideration 10 ≤ C ≤ 20. EXAMPLE 9 Finding a Fahrenheit Temperature from a Celsius Range © 2010 Pearson Education, Inc. All rights reserved 29

Solution continued So, the temperature range from 10º to 20º Celsius corresponds to a range from 50º to 68º Fahrenheit. EXAMPLE 9 Finding a Fahrenheit Temperature from a Celsius Range © 2010 Pearson Education, Inc. All rights reserved 30