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Chapter 1.8 Absolute Value and Inequalities Recall from Chapter R that the absolute value of a number a, written |a|, gives the distance from a to 0 on a number line. By this definition, the equation |x| = 3 can be solved by finding all real numbers at a distance of 3 units from 0. |x| = 3 Two numbers satisfy this equation, 3 and -3, so the solution set is {-3, 3}. Distance Distance is 3. Distance is is 3. Distance is less than 3. Distance is greater than 3. greater than 3. | -3 | 0 | 3 Similarly |x| < 3 is satisfied by all real numbers whose distances from 0 are less than 3, that is, the interval -3<x<3 or (-3,3). Finally x 3 is satisfied by all real numbers whose distances from 0 are greater th an 3. So the solution set is - ,-3 3, . Properties of Absolute Value 1. For b 0, a b if and only if a b or a - b 2. a b if and only if a b or a - b 3. a b if and only if -ba b 4. a b if and only if a b or a b Example 1 Solving Absolute Value Equations Solve 5 - 3x 12 Example 1 Solving Absolute Value Equations Solve 4x - 3 x 6 Example 2 Solving Absolute Value Inequalities Solve 2x 1 7 Example 2 Solving Absolute Value Inequalities Solve 2x 1 7 Example 3 Solving Absolute Value Inequalities Requiring a Transformation Solve 2 - 7x 1 4 Example 4 Solving Absolute Value Equations and Inequalities Solve 2 5x 4 Example 4 Solving Absolute Value Equations and Inequalities Solve 4x 7 3 Absolute Value Models for Distance and Tolerance Recall from Selection R.2 that is a and b represent two real numbers, then the absolute value of their difference, either |a - b| or |b – a|, represent the distance between them. This fact is used to write absolute value equations or inequalities to express distance. Example 5 Using Absolute Value Inequaliteis to Describe Distances Write each statement using an absolute value inequality. (a) k is no less than 5 units from 8 Example 5 Using Absolute Value Inequaliteis to Describe Distances Write each statement using an absolute value inequality. (b) n is within .001 unit of 6 Example 6 Using Absolute Value to Model Tolerance Suppose y = 2x + 1 and we want y to be within .01 unit of 4. For what values of x will this be true? |y – 4| < .01 |2x + 1 – 4| < .01 Homework Section 1.8 # 1 - 82