Solving Absolute Value Inequalities Worksheet Answers

Solving Absolute Value Inequalities: A Comprehensive Guide with Worksheet Answers

Absolute value inequalities can seem daunting at first, but with a systematic approach and a solid understanding of the underlying principles, you can master them. This comprehensive guide will walk you through the process of solving absolute value inequalities, providing clear explanations, examples, and – most importantly – the answers to a practice worksheet. We'll cover various scenarios and techniques to ensure you're well-equipped to tackle any absolute value inequality problem.

Understanding Absolute Value

Before diving into inequalities, let's refresh our understanding of absolute value. The absolute value of a number is its distance from zero on the number line. It's always non-negative. We denote the absolute value of a number x as |x|.

• |x| = x if x ≥ 0 (e.g., |5| = 5)
• |x| = -x if x < 0 (e.g., |-5| = 5)

This seemingly simple definition is key to understanding how to solve absolute value inequalities.

Solving Absolute Value Inequalities: The Core Principles

Solving absolute value inequalities involves manipulating the inequality to isolate the absolute value expression. The key is to remember the two scenarios that arise:

1. |x| < a: This means that x is less than a units away from zero. This translates to -a < x < a. This is a compound inequality.

2. |x| > a: This means that x is more than a units away from zero. This translates to x < -a or x > a. This is a disjunction (or a union of two inequalities).

These principles are fundamental. Let's illustrate with examples:

Example 1: |x| < 3

This inequality means that x is less than 3 units away from 0. Therefore, the solution is -3 < x < 3. This is an interval represented as (-3, 3) in interval notation.

Example 2: |x| > 2

This inequality means that x is more than 2 units away from 0. This gives us two separate inequalities: x < -2 or x > 2. In interval notation, this is (-∞, -2) ∪ (2, ∞).

Solving More Complex Absolute Value Inequalities

Many absolute value inequalities are more complex than these basic examples. They often involve expressions inside the absolute value symbols and might require multiple steps to solve. Let's explore strategies for tackling these more challenging problems.

Example 3: |2x + 1| ≤ 5

1. Isolate the absolute value: The absolute value expression is already isolated.

2. Apply the principle for |x| < a: This inequality is equivalent to -5 ≤ 2x + 1 ≤ 5.

3. Solve the compound inequality:

   • Subtract 1 from all parts: -6 ≤ 2x ≤ 4
   • Divide all parts by 2: -3 ≤ x ≤ 2

The solution is [-3, 2].

Example 4: |3x - 2| > 4

1. Isolate the absolute value: The absolute value expression is already isolated.

2. Apply the principle for |x| > a: This inequality is equivalent to 3x - 2 < -4 or 3x - 2 > 4.

3. Solve each inequality separately:

   • 3x - 2 < -4 => 3x < -2 => x < -2/3
   • 3x - 2 > 4 => 3x > 6 => x > 2

The solution is (-∞, -2/3) ∪ (2, ∞).

Dealing with Inequalities Involving Absolute Value and Other Operations

When dealing with absolute value inequalities that include addition, subtraction, multiplication, or division, always remember the order of operations (PEMDAS/BODMAS). Solve for the absolute value term first, then proceed as outlined above.

Example 5: 2|x - 4| + 3 > 7

1. Isolate the absolute value:

   • Subtract 3 from both sides: 2|x - 4| > 4
   • Divide both sides by 2: |x - 4| > 2

2. Apply the principle for |x| > a: x - 4 < -2 or x - 4 > 2

3. Solve each inequality separately:

   • x - 4 < -2 => x < 2
   • x - 4 > 2 => x > 6

The solution is (-∞, 2) ∪ (6, ∞).

Absolute Value Inequalities: Worksheet and Answers

Now let's put your skills to the test with a worksheet. Remember to follow the steps outlined above: isolate the absolute value, apply the appropriate principle, and solve the resulting inequalities.

Worksheet:

1. |x| ≤ 7
2. |x - 1| > 3
3. |2x + 5| < 9
4. |3x - 4| ≥ 2
5. |x + 2| + 1 < 5
6. 3|x - 1| - 2 ≥ 7
7. -2|x + 3| + 4 ≥ -2
8. |4x + 1| - 3 ≤ 5
9. |2 - 5x| > 8
10. |1/2 x - 3| ≤ 1

Answers:

1. [-7, 7]
2. (-∞, -2) ∪ (4, ∞)
3. (-7, 2)
4. (-∞, 2/3] ∪ [2, ∞)
5. (-6, 2)
6. (-∞, -2] ∪ [4, ∞)
7. [-6, 0]
8. [-9/2, 7/2]
9. (-∞, -6/5) ∪ (2, ∞)
10. [4, 8]

Troubleshooting Common Mistakes

Many students struggle with absolute value inequalities due to a few common errors:

• Forgetting the two cases: Remember, |x| > a leads to two separate inequalities, not a single one.

• Incorrectly handling compound inequalities: When dealing with -a < x < a, maintain the order and perform operations on all three parts consistently.

• Errors in algebraic manipulation: Double-check your algebra steps to ensure accuracy. A single mistake can invalidate the entire solution.

• Neglecting to consider the sign: Always remember that absolute value is always non-negative.

• Improper interval notation: Ensure your interval notation correctly represents the solution sets.

Advanced Techniques and Applications

While the principles discussed above cover most absolute value inequalities encountered in introductory algebra, more advanced techniques exist for more complex scenarios, including those involving multiple absolute value terms or those requiring the use of graphing calculators for visualization and solution verification. These more advanced techniques may involve piecewise functions or the analysis of critical points.

Conclusion

Solving absolute value inequalities requires a methodical approach and a firm understanding of the core principles. By consistently applying these techniques and carefully checking your work, you can successfully navigate these types of problems. Remember to break down complex problems into smaller, manageable steps and practice regularly to build your confidence and proficiency. This comprehensive guide, complete with a worksheet and answers, provides a strong foundation for mastering this important algebraic concept. Remember to review the common mistakes section to identify and avoid pitfalls. Consistent practice will lead to mastery, and soon solving absolute value inequalities will be second nature.