solving absolute value inequalities worksheet

Guri Bee

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22-10-2024

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Mastering Absolute Value Inequalities: A Comprehensive Guide

Absolute value inequalities can seem daunting at first, but with the right approach, they become manageable. This article will guide you through the process of solving these inequalities, using examples and insights gleaned from GitHub discussions.

Understanding Absolute Value

Before diving into inequalities, let's recap what absolute value means. The absolute value of a number represents its distance from zero on the number line. For example, |3| = 3 and |-3| = 3.

Solving Absolute Value Inequalities

The key to solving absolute value inequalities lies in recognizing that the absolute value of an expression can be either positive or negative. We need to consider both possibilities.

Case 1: The expression inside the absolute value is positive.

• Example: Solve the inequality |x - 2| < 5

  1. Consider the positive case: x - 2 < 5
  2. Solve for x: x < 7

• Case 2: The expression inside the absolute value is negative.

  1. Consider the negative case: -(x - 2) < 5
  2. Simplify: -x + 2 < 5
  3. Solve for x: -x < 3
  4. Multiply both sides by -1 (and flip the inequality sign): x > -3

• Combining both cases: We found that x < 7 and x > -3. This means the solution is -3 < x < 7.

Practical Example (Inspired by GitHub Discussion)

Let's look at a practical example from a GitHub discussion [link to relevant GitHub discussion]. A user asked how to solve the inequality |2x - 1| > 3. Following our approach:

1. Positive Case: 2x - 1 > 3 => 2x > 4 => x > 2

2. Negative Case: -(2x - 1) > 3 => -2x + 1 > 3 => -2x > 2 => x < -1

3. Solution: x < -1 or x > 2

Additional Tips and Tricks

• Graphing: Visualizing the solutions on a number line can be helpful. For example, in the previous example, you'd shade everything to the left of -1 and everything to the right of 2.
• Compound Inequalities: Inequalities may require more than one step. For example, |x - 3| < 2 and |2x + 1| > 5 can be solved by applying the case-by-case method to both inequalities.

Conclusion

Solving absolute value inequalities requires careful attention to both the positive and negative cases of the expression within the absolute value. By understanding these concepts and following the steps outlined in this article, you can confidently tackle even complex inequalities. Remember to check your solutions and utilize resources like GitHub discussions for further clarification or additional examples.