Section 2.5: Absolute Value Functions and Inequalities
This comprehensive section covers the fundamentals of absolute value functions, their graphing techniques, and solving related equations and inequalities. The material begins with basic absolute value function y = |x| and progresses to more complex examples involving vertical stretches and translations.
Definition: An absolute value function represents the distance of a number from zero on a number line.
Example: For the function y = 2|x-3|-1, the graph involves a vertical stretch factor of 2, with shifts right 3 and down 1.
Highlight: When solving absolute value equations, remember that |x| = 5 means x = 5 or x = -5, as absolute values can never be negative.
Example: In solving |5x-7| = 10:
- First equation: 5x-7 = 10
- Second equation: 5x-7 = -10
- Solutions: x = 17/5 or x = -3
Vocabulary: Domain and range restrictions are crucial when dealing with absolute value inequalities.
Example: For |x| ≤ 3:
- This means the distance x is from 0 has to be less than or equal to 3
- Solution interval:
The section concludes with more complex inequality examples, including compound inequalities and their graphical representations on a number line.
