Chapter 4: Solving Inequalities

4.6Absolute 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 or 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| ≥ 6 and 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 0.008 mm. Find the range of acceptable diameters for the piston.

Example 4a

• The ideal weight of one type of model airplane engine is 33.86 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. 237

• 2-20 even, 28, 34, 38, 44, 50, 79