2. Copyright © 2008 Pearson Education, Inc. CHAPTER 2: Functions, Equations, and Inequalities 2.1 Linear Equations, Functions, and Models 2.2 The Complex Numbers 2.3 Quadratic Equations, Functions, and Models 2.4 Analyzing Graphs of Quadratic Functions 2.5 More Equation Solving 2.6 Solving Linear Inequalities

3. Copyright © 2008 Pearson Education, Inc. 2.6 Solving Linear Inequalities  Solve linear inequalities.  Solve compound inequalities.  Solve inequalities with absolute value.  Solve applied problems using inequalities.

4. Slide 2.6-4 Copyright © 2008 Pearson Education, Inc. Inequalities  An inequality is a sentence with, , or  as its verb. Examples: 5x  7 < 3 + 4x 3(x + 6)   4(x  3)

5. Slide 2.6-5 Copyright © 2008 Pearson Education, Inc. Principles for Solving Inequalities For any real numbers a, b, and c: The Addition Principle for Inequalities: If a < b is true, then a + c < b + c is true. The Multiplication Principle for Inequalities: If a 0 are true, then ac bc is true. Similar statements hold for a  b. When both sides of an inequality are multiplied or divided by a negative number, we must reverse the inequality sign.

6. Slide 2.6-6 Copyright © 2008 Pearson Education, Inc. Examples Solve: {x|x <  2} or ( ,  2) Solve: {x|x   4} or [  4,  ) 0–55 ) 0 5 [

7. Slide 2.6-7 Copyright © 2008 Pearson Education, Inc. Compound Inequalities When two inequalities are joined by the word and or the word or, a compound inequality is formed. Conjunction contains the word and. Example:  7 < 3x + 5 and 3x + 9  6 Disjunction contains the word or. Example: 3x + 5  6 or 3x + 6 > 12

8. Slide 2.6-8 Copyright © 2008 Pearson Education, Inc. Examples Solve: 4x  5   3 or 4x  5 > 3 0–55 ( ] 0 5 ( ]

9. Slide 2.6-9 Copyright © 2008 Pearson Education, Inc. Inequalities with Absolute Value Inequalities sometimes contain absolute-value notation. The following properties are used to solve them. For a > 0 and an algebraic expression X: |X| < a is equivalent to  a < X < a. |X| > a is equivalent to X a. Similar statements hold for |X|  a and |X|  a.

10. Slide 2.6-10 Copyright © 2008 Pearson Education, Inc. Example  Solve: 0–55 ()

11. Slide 2.6-11 Copyright © 2008 Pearson Education, Inc. Application Johnson Catering charges $100 plus $30 per hour to cater an event. Catherine’s Catering charges $50 per hour. For what lengths of time does it cost less to hire Catherine’s Catering? 1. Familiarize. Read the problem. 2. Translate. Catherine’s is less than Johnson 50x < 100 + 30x

12. Slide 2.6-12 Copyright © 2008 Pearson Education, Inc. Application continued 3. Carry out. 4. Check. 5. State. For values of x < 5 hr, Catherine’s Catering will cost less.