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Understanding and Solving Compound Inequalities: A Comprehensive Guide

This guide explores compound inequalities, defined as two separate inequalities connected by "and" or "or." It clarifies how to graph these inequalities, noting that "and" leads to intersection (graphs towards each other) while "or" leads to union (graphs away from each other). Examples illustrate how to write and graph compound inequalities, as well as solve them step-by-step. The guide also includes practice problems to solidify understanding, making it an essential resource for mastering this topic in mathematics.

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Understanding and Solving Compound Inequalities: A Comprehensive Guide

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  1. Section 6-4 Solve Compound Inequalities

  2. Compound Inequality: Two separate inequalities joined by and or or. • The graph of a compound inequality with and is the intersection of the graphs of the inequalities. • The graphs go TOWARDS each other. • The graph of a compound inequality with or is the union of the graphs of the inequalities. • The graphs go AWAY from each other.

  3. Example 1 Write and graph compound inequalities: a) All real numbers that are greater than or equal to -4 and less than 4. -4 ≤ x < 4 Graph: -8 -6 -4 -2 0 2 4 6

  4. Example 1 Continued Write and graph compound inequalities: b) All real numbers that are less than -1 or greater than 2. x < -1 or x > 2 Graph: -3 -2 -1 0 1 2 3 4

  5. Example 2 Solve a compound inequality with and: 1 < -2x + 3 < 19 -3 - 3 - 3 -2 < -2x < 16 -2 -2 -2 1 > x > -8 -8 < x < 1 Graph: -8 -6 -4 -2 0 2 4 6 Given APOI Simplify Flip inequalities b/c divided by negative! MPOI Simplify

  6. Example 3 Solve each inequality separately. Solve a compound inequality with or: 3x – 2 ≤ -11or 2x + 8 > 16 Graph: -8 -6 -4 -2 0 2 4 6 Given Given +2 +2 - 8 -8 APOI APOI Simplify 3x ≤ -9 2x > 8 Simplify 3 3 2 2 MPOI MPOI Simplify Simplify or x > 4 x ≤ -3

  7. Homework Section 6-4 Pg. 384 – 387 3 – 5 Solve & Prove: 9 – 19 odd Follow Directions: 21

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