Equations and Inequalities

1. Chapter 2 Equations and Inequalities

2. Chapter Sections 2.1 – Solving Linear Equations 2.2 – Problem Solving and Using Formulas 2.3 – Applications of Algebra 2.4 – Additional Application Problems 2.5 – Solving Linear Inequalities 2.6 – Solving Equations and Inequalities Containing Absolute Values

3. Solving Equations and Inequalities Containing Absolute Values § 2.6

4. 3 units 3 units -5 -4 -3 -2 -1 0 1 2 3 4 5 Absolute Value The absolutevalue of a number x, symbolized |x|, is the distance x is from 0 on the number line. The absolute value of every number will be either 0 or positive. |3| = 3 and |-3| = 3

5. Absolute Value Equations To Solve Inequalities of the Form |x| = a, a > 0 If |x| = a and a > 0, then x = a or x = -a. Example: Solve the inequality |x| = 2 Using the procedure, we get x = 2 or x = -2. The solution set is {-2, 2}

6. Absolute Value Equations To Solve Inequalities of the Form |x| < a If |x| < a and a > 0, then –a < x < a. Example: Solve the inequality |2x – 3| < 5 The solution set is {x| -1 < x < 4}.

7. Absolute Value Equations If |x| >a and a > 0, then x < –a or x > a. To Solve Inequalities of the Form |x| > a Example: Solve the inequality |2x – 3| > 5 or The solution set is {x| x < -1 or x > 4}

8. Absolute Value Equations To Solve Inequalities of the Form |x| < a or |x| > a If |x| < a and a > 0, then –a < x < a. Example: Solve the inequality |6x – 8| + 5 < 3 Since |6x – 8| will always be greater than or equal to 0 for any real number x, this inequality can never be true. Therefore, the solution is the empty set, ᴓ.

9. Absolute Value Equations To Solve Inequalities of the Form |x| < 0, |x|  0, |x| > 0, or |x| ≥ 0

10. Absolute Value Equations If |x| = |y|, then x = y or x = y. To Solve Inequalities of the Form |x| = |y| Example: Solve the inequality |z + 3| = |2z - 7| or The solution set is {10, 4/3}.