1. Slide 1.6- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Inequalities Learn the vocabulary for discussing inequalities. Learn to solve and graph linear inequalities. Learn to solve and graph a combined inequality. Learn to solve and graph an inequality involving the reciprocal of a linear expression. SECTION 1.6 1 2 3 4

3. Slide 1.6- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

4. Slide 1.6- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

5. Slide 1.6- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

6. Slide 1.6- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

7. Slide 1.6- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE 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.

8. Slide 1.6- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

9. Slide 1.6- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sign of C InequalitySenseExample A < BA < B<3x < 12 positiveAC < B CUnchanged positiveUnchanged negativeAC > B CReversed negativeReversed

10. Slide 1.6- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

11. Slide 1.6- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

12. Slide 1.6- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solving and Graphing Linear Inequalities The solution set is {x|x ≥ 2}, or [2, ∞). Solution continued [ 120

13. Slide 1.6- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

14. Slide 1.6- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Let t = time elapsed since plane on autopilot 300t = distance plane flown in t hours 150 + 300t = 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 ≥

15. Slide 1.6- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

16. Slide 1.6- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solving & Graphing a Compound Inequality Solve the inequality graph its solution set. and Solution First, solve the inequalities separately.

17. Slide 1.6- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 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

18. Slide 1.6- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE RECIPROCAL SIGN PROPERTY If x ≠ 0, x andare either both positive or negative. In symbols, if x > 0, then and if x < 0, then

19. Slide 1.6- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 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, ∞). [ 120453

20. Slide 1.6- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 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.

21. Slide 1.6- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 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

22. Slide 1.6- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding a Fahrenheit Temperature from a Celsius Range Let C = temperature in Celsius degrees. Solution For the three weeks under consideration 10 ≤ C ≤ 20.

23. Slide 1.6- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding a Fahrenheit Temperature from a Celsius Range Solution continued So, the temperature range from 10º to 20º Celsius corresponds to a range from 50º to 68º Fahrenheit.