How To Solve An Inequality With Absolute Value

Solving inequalities involving absolute values might seem daunting at first, but with a step-by-step approach and a clear understanding of the underlying principles, you can master this skill. This article will guide you through the process, providing explanations, examples, and tips to help you confidently solve any absolute value inequality.

Understanding Absolute Value

Before diving into inequalities, it's crucial to understand what absolute value means. The absolute value of a number, denoted as |x|, represents its distance from zero on the number line. Distance is always non-negative, so |x| is always greater than or equal to zero, regardless of whether x is positive or negative.

• For example:
  • |3| = 3 (3 is 3 units away from zero)
  • |-3| = 3 (-3 is also 3 units away from zero)
  • |0| = 0 (0 is 0 units away from zero)

This concept of distance is fundamental to understanding how to solve absolute value inequalities.

The Two Cases of Absolute Value

The key to solving absolute value inequalities lies in recognizing that the expression inside the absolute value can be either positive or negative. This gives rise to two separate cases that need to be considered:

• Case 1: The expression inside the absolute value is non-negative. In this case, the absolute value doesn't change the expression. So, |x| = x.
• Case 2: The expression inside the absolute value is negative. In this case, the absolute value makes the expression positive by multiplying it by -1. So, |x| = -x.

Solving Absolute Value Inequalities: A Step-by-Step Guide

Now, let's break down the process of solving absolute value inequalities into manageable steps. We'll use examples to illustrate each step.

Step 1: Isolate the Absolute Value Expression

Before you can start splitting the inequality into cases, you need to isolate the absolute value expression on one side of the inequality. This means getting rid of any constants or coefficients that are outside the absolute value bars.

• Example 1: Solve |2x + 1| - 3 < 2

  • Add 3 to both sides: |2x + 1| < 5

• Example 2: Solve 3|x - 4| + 5 ≥ 14

  • Subtract 5 from both sides: 3|x - 4| ≥ 9
  • Divide both sides by 3: |x - 4| ≥ 3

Step 2: Split the Inequality into Two Cases

Once the absolute value expression is isolated, you'll create two separate inequalities, one for each case:

• Case 1: The expression inside the absolute value is non-negative. In this case, simply remove the absolute value bars and keep the inequality sign the same.

• Case 2: The expression inside the absolute value is negative. In this case, remove the absolute value bars, reverse the inequality sign, and multiply the expression on the other side of the inequality by -1.

• Example 1 (continued): |2x + 1| < 5

  • Case 1: 2x + 1 < 5
  • Case 2: 2x + 1 > -5 (Notice the inequality sign is reversed)

• Example 2 (continued): |x - 4| ≥ 3

  • Case 1: x - 4 ≥ 3
  • Case 2: x - 4 ≤ -3 (Notice the inequality sign is reversed)

Step 3: Solve Each Inequality Separately

Now you have two separate inequalities that you can solve using standard algebraic techniques.

• Example 1 (continued):

  • Case 1: 2x + 1 < 5
    • Subtract 1 from both sides: 2x < 4
    • Divide both sides by 2: x < 2
  • Case 2: 2x + 1 > -5
    • Subtract 1 from both sides: 2x > -6
    • Divide both sides by 2: x > -3

• Example 2 (continued):

  • Case 1: x - 4 ≥ 3
    • Add 4 to both sides: x ≥ 7
  • Case 2: x - 4 ≤ -3
    • Add 4 to both sides: x ≤ 1

Step 4: Determine the Solution Set

The solution set depends on the original inequality sign. There are two possibilities:

• If the original inequality was of the form |expression| < constant or |expression| ≤ constant ("less than" or "less than or equal to"): The solution is the intersection of the solutions from Case 1 and Case 2. This means you're looking for the values of x that satisfy both inequalities. This is often expressed as a compound inequality of the form a < x < b or a ≤ x ≤ b.

• If the original inequality was of the form |expression| > constant or |expression| ≥ constant ("greater than" or "greater than or equal to"): The solution is the union of the solutions from Case 1 and Case 2. This means you're looking for the values of x that satisfy either inequality. This is often expressed as two separate inequalities connected by the word "or".

• Example 1 (continued): The original inequality was |2x + 1| < 5 ("less than"). Therefore, we need the intersection of x < 2 and x > -3. This can be written as the compound inequality: -3 < x < 2. In interval notation, this is (-3, 2).

• Example 2 (continued): The original inequality was |x - 4| ≥ 3 ("greater than or equal to"). Therefore, we need the union of x ≥ 7 and x ≤ 1. This is written as: x ≤ 1 or x ≥ 7. In interval notation, this is (-∞, 1] ∪ [7, ∞).

Step 5: Check Your Solution (Optional but Recommended)

To ensure you haven't made any errors, it's always a good idea to check your solution by plugging in values from your solution set back into the original inequality. Choose values that are within the solution set and values that are outside the solution set to confirm that your solution is correct.

Detailed Examples with Explanations

Let's work through some more examples to solidify your understanding.

Example 3: Solve |3x - 6| ≤ 9

1. Isolate the absolute value: The absolute value is already isolated.
2. Split into two cases:
   • Case 1: 3x - 6 ≤ 9
   • Case 2: 3x - 6 ≥ -9
3. Solve each inequality:
   • Case 1:
     • Add 6 to both sides: 3x ≤ 15
     • Divide both sides by 3: x ≤ 5
   • Case 2:
     • Add 6 to both sides: 3x ≥ -3
     • Divide both sides by 3: x ≥ -1
4. Determine the solution set: Since the original inequality was "less than or equal to," we need the intersection of x ≤ 5 and x ≥ -1. This is -1 ≤ x ≤ 5. In interval notation, this is [-1, 5].
5. Check your solution (optional):
   • Let's try x = 0 (which is within the solution set): |3(0) - 6| = |-6| = 6 ≤ 9. This is true.
   • Let's try x = 6 (which is outside the solution set): |3(6) - 6| = |12| = 12 ≤ 9. This is false.

Example 4: Solve |(1/2)x + 2| > 4

1. Isolate the absolute value: The absolute value is already isolated.
2. Split into two cases:
   • Case 1: (1/2)x + 2 > 4
   • Case 2: (1/2)x + 2 < -4
3. Solve each inequality:
   • Case 1:
     • Subtract 2 from both sides: (1/2)x > 2
     • Multiply both sides by 2: x > 4
   • Case 2:
     • Subtract 2 from both sides: (1/2)x < -6
     • Multiply both sides by 2: x < -12
4. Determine the solution set: Since the original inequality was "greater than," we need the union of x > 4 and x < -12. This is x < -12 or x > 4. In interval notation, this is (-∞, -12) ∪ (4, ∞).
5. Check your solution (optional):
   • Let's try x = 5 (which is within the solution set): |(1/2)(5) + 2| = |2.5 + 2| = |4.5| = 4.5 > 4. This is true.
   • Let's try x = -13 (which is within the solution set): |(1/2)(-13) + 2| = |-6.5 + 2| = |-4.5| = 4.5 > 4. This is true.
   • Let's try x = 0 (which is outside the solution set): |(1/2)(0) + 2| = |2| = 2 > 4. This is false.

Example 5: Solve |5 - 2x| ≥ 1

1. Isolate the absolute value: The absolute value is already isolated.
2. Split into two cases:
   • Case 1: 5 - 2x ≥ 1
   • Case 2: 5 - 2x ≤ -1
3. Solve each inequality:
   • Case 1:
     • Subtract 5 from both sides: -2x ≥ -4
     • Divide both sides by -2 (and reverse the inequality sign!): x ≤ 2
   • Case 2:
     • Subtract 5 from both sides: -2x ≤ -6
     • Divide both sides by -2 (and reverse the inequality sign!): x ≥ 3
4. Determine the solution set: Since the original inequality was "greater than or equal to," we need the union of x ≤ 2 and x ≥ 3. This is x ≤ 2 or x ≥ 3. In interval notation, this is (-∞, 2] ∪ [3, ∞).

Example 6: Solve |4x + 8| < 0

This is a special case. Recall that the absolute value of any expression is always non-negative (greater than or equal to zero). Therefore, the absolute value of an expression can never be strictly less than zero. This inequality has no solution.

Example 7: Solve |x - 3| + 5 ≤ 5

1. Isolate the absolute value: Subtract 5 from both sides: |x - 3| ≤ 0
2. Analyze the inequality: The absolute value of an expression is always non-negative. Therefore, |x - 3| can be equal to zero, but it can never be less than zero. So, the only way for this inequality to be true is if |x - 3| = 0.
3. Solve for x: If |x - 3| = 0, then x - 3 = 0, which means x = 3.
4. Solution Set: The solution is x = 3.

Common Mistakes to Avoid

• Forgetting to split into two cases: This is the most common mistake. Remember that the expression inside the absolute value can be either positive or negative.
• Forgetting to reverse the inequality sign: When dealing with the negative case, remember to reverse the inequality sign.
• Not isolating the absolute value first: You must isolate the absolute value expression before splitting into cases.
• Incorrectly determining the solution set: Remember to use the intersection for "less than" inequalities and the union for "greater than" inequalities.
• Dividing or multiplying by a negative number without reversing the inequality sign: This applies to both the absolute value cases and the individual inequalities you solve.

Advanced Techniques and Special Cases

• Absolute Value on Both Sides: If you have absolute value expressions on both sides of the inequality, you can still split it into cases, but you'll need to consider more scenarios. A common technique is to square both sides of the inequality to eliminate the absolute value signs. However, be careful when squaring inequalities, as it can introduce extraneous solutions. You must check your solutions in the original inequality.

  • Example: |x + 1| < |2x - 1|
    • Squaring both sides: (x + 1)² < (2x - 1)²
    • Expanding: x² + 2x + 1 < 4x² - 4x + 1
    • Simplifying: 0 < 3x² - 6x
    • Factoring: 0 < 3x(x - 2)
    • Solving the quadratic inequality: x < 0 or x > 2. You would then need to verify these solutions in the original absolute value inequality.

• Nested Absolute Values: If you have absolute values inside other absolute values, work from the innermost absolute value outwards.

• No Solution or All Real Numbers: Be aware that some absolute value inequalities may have no solution or may be true for all real numbers. This often happens when the absolute value expression is compared to a negative number or when the inequality simplifies to a statement that is always true or always false.

The Importance of Understanding Absolute Value Inequalities

Understanding and being able to solve absolute value inequalities is important for several reasons:

• Foundation for Higher-Level Math: Absolute value inequalities are a fundamental concept in algebra and calculus. They appear in more advanced topics such as limits, continuity, and optimization.
• Problem-Solving Skills: Solving absolute value inequalities requires logical thinking, careful attention to detail, and the ability to break down complex problems into simpler steps. These skills are valuable in many areas of life.
• Real-World Applications: Absolute values are used to represent distance, error, and tolerance in various real-world applications, such as engineering, physics, and economics. Understanding absolute value inequalities can help you solve problems related to these applications.
• Standardized Tests: Absolute value inequalities are often tested on standardized tests such as the SAT and ACT. Mastering this topic can improve your score and increase your chances of getting into your desired college or university.

Conclusion

Solving absolute value inequalities requires a systematic approach and a clear understanding of the properties of absolute values. By following the steps outlined in this article, practicing with examples, and avoiding common mistakes, you can confidently solve any absolute value inequality you encounter. Remember to always isolate the absolute value expression, split the inequality into two cases, solve each case separately, and carefully determine the solution set based on the original inequality sign. With practice, you'll become proficient at solving these types of problems and gain a deeper appreciation for the power and versatility of absolute values in mathematics.