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

OBJECTIVES Equations & Inequalities Involving Absolute Value Solve equations involving absolute value. Solve inequalities involving absolute value. SECTION © 2010 Pearson Education, Inc. All rights reserved

ABSOLUTE VALUE © 2010 Pearson Education, Inc. All rights reserved 3

THE SOLUTIONS OF If a ≥ 0 and u is an algebraic expression, then © 2010 Pearson Education, Inc. All rights reserved 4

EXAMPLE 1 Solving an Equation Involving Absolute Value Solve each equation. The solution set is {–3}. Solution Check the solution. © 2010 Pearson Education, Inc. All rights reserved 5

EXAMPLE 1 Solving an Equation Involving Absolute Value The solution set is {–5, 8}. Solution continued We leave it to you to check the solutions. Isolate the absolute value. © 2010 Pearson Education, Inc. All rights reserved 6

EXAMPLE 2 Solving an Equation of the Form If |u| = |v|, then either u is equal to |v| or u is equal to –|v|. Since |v| = ± v in every case, we have u = v or u = –v. Thus, Solution Solve The solution set is {–2}. We leave the check to you. © 2010 Pearson Education, Inc. All rights reserved 7

EXAMPLE 3 Solving an Equation of the Form Letting u = 2x + 1 and v = x + 5, the equation is equivalent to Solution Solve © 2010 Pearson Education, Inc. All rights reserved 8

EXAMPLE 3 Solving an Equation of the Form Check: Solution continued or (see next slide) © 2010 Pearson Education, Inc. All rights reserved 9

EXAMPLE 3 Solving an Equation of the Form Check: Solution continued The solution set is x = 6 or © 2010 Pearson Education, Inc. All rights reserved 10

RULES FOR SOLVING ABSOLUTE VALUE INEQUALITIES If a > 0, and u is an algebraic expression, then © 2010 Pearson Education, Inc. All rights reserved 11

EXAMPLE 4 Solving an Inequality Involving an Absolute Value Solve the inequalityand graph the the solution set. Rule 2 applies here, with u = 4x – 1 and a = 9. Solution © 2010 Pearson Education, Inc. All rights reserved 12

EXAMPLE 4 Solving an Inequality Involving an Absolute Value Solution continued The solution set isthat is, the solution set is the closed interval 120–1–2 ] [ 3–3 © 2010 Pearson Education, Inc. All rights reserved 13

EXAMPLE 5 We want to find the possible search range (in miles) for a search plane that has 30 gallons of fuel and uses 10 gallons of fuel per hour. The search plane normally averages 110 miles per hour, but that weather conditions could affect the average speed by as much as 15 miles per hour (either slower or faster). How do we find the possible search range? Finding the Search Range of an Aircraft © 2010 Pearson Education, Inc. All rights reserved 14

EXAMPLE 5 To find distance, we need both time and speed. Let x = actual speed in mph We know actual speed is within 15 mph of average speed, 110 mph. That is, |actual speed – average speed| ≤ 15 mph Finding the Search Range of an Aircraft Solution The actual speed of the search plane is between 95 and 125 mph. © 2010 Pearson Education, Inc. All rights reserved 15

EXAMPLE 5 The plane uses 10 g of fuel per hour. It has 30 g, so it can fly for 3 hr. So the actual number of miles the search plane can fly is 3x. Finding the Search Range of an Aircraft Solution continued The search plane’s range is between 285 and 375 miles. © 2010 Pearson Education, Inc. All rights reserved 16

EXAMPLE 6 Solving an Inequality Involving an Absolute Value Solve the inequalityand graph the the solution set. Rule 4 applies here, with u = 2x – 8 and a = 9. Solution © 2010 Pearson Education, Inc. All rights reserved 17

EXAMPLE 6 Solving an Inequality Involving an Absolute Value The solution set is {x | x ≤ 2 or x ≥ 6}; Solution continued x ≤ 2 or x ≥ 6 (–∞, 2] U [6, ∞) [] in interval notation it is (–∞, 2] U [6, ∞). © 2010 Pearson Education, Inc. All rights reserved 18

EXAMPLE 7 Solving Special Cases of Absolute Value Inequalities Solve each inequality. a. The absolute value is always nonnegative, so |3x – 2| > –5 is true for all real numbers x. The solution set is all real numbers, or (−∞, ∞). Solution b. There is no real number with absolute value ≤ –2. The solution set is the empty set, or © 2010 Pearson Education, Inc. All rights reserved 19