Chapter 4: Solving Inequalities 4.6 Absolute Value Equations and Inequalities

Absolute Value Distance from a number to zero Always positive! – (because distance is never negative) Looks like:| x |

Example 1 Solve|x| + 5 = 11

Example 1a Solve|t| - 2 = -1

Example 1b Solve3|n| = 15

Example 1c Solve4 = 3|w| - 2

Example 1d Is there a solution of 2|n| = -15?

Example 2 Solve|2p + 5| = 11

Example 2a Solve|c – 2| = 6

Example 2b Solve-5.5 = |t + 2|

Example 2c Solve|7d| = 14

Solving Absolute Value Equations To solve an equation in the form |A| = b, where A represents a variable expression and b > 0, solve A = b and A = -b In other words, isolate the absolute value part, then set it equal to the positive and the negative of the right side

Solving Absolute Value Inequalities For |A| < b (think “less - and”) – Solve –b < A < b For |A| > b (think “great – or”) – Solve A b

Example 3 Solve|v – 3| ≥ 4 and graph the solutions

Example 3a Solve|w + 2| > 5 and graph the solutions

Example 3b Solve |3d| ≥ 6and graph the solutions

Example 3c Solve9 < |c + 7| and graph the solutions

Example 3d Solve4 – 3|m + 2| > -14 and graph the solutions

Example 4 The ideal diameter of a piston for one type of car engine is 90,000 mm. The actual diameter can vary from the ideal by at most mm. Find the range of acceptable diameters for the piston.

Example 4a The ideal weight of one type of model airplane engine is ounces. The actual weight may vary from the ideal by at most 0.05 ounces. Find the range of acceptable weights for this engine.

Homework P even, 28, 34, 38, 44, 50, 79