How To Solve Inequalities With Absolute Values On Both Sides

How to Solve Inequalities with Absolute Values on Both Sides

Solving inequalities involving absolute values on both sides can seem daunting, but with a systematic approach and a solid understanding of absolute value properties, you can master this skill. This comprehensive guide will walk you through various scenarios, providing clear explanations and examples to help you confidently tackle these types of problems.

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. Therefore, it's always non-negative. Mathematically:

• |x| = x if x ≥ 0
• |x| = -x if x < 0

This means |5| = 5 and |-5| = 5. The absolute value "strips away" the negative sign.

Solving Inequalities with Absolute Values: The Core Principles

When dealing with inequalities containing absolute values on both sides, the core strategy involves considering different cases based on the potential signs of the expressions inside the absolute value symbols. We'll analyze these cases systematically.

Let's consider a general inequality of the form:

|A| ≤ |B| or |A| ≥ |B|

where A and B are algebraic expressions.

Case 1: |A| ≤ |B|

This inequality implies that the absolute value of A is less than or equal to the absolute value of B. This means A is within the range of -B to B, inclusive. We can represent this as a compound inequality:

-B ≤ A ≤ B

To solve, we need to address three distinct cases:

1. A and B are both non-negative: If A ≥ 0 and B ≥ 0, we simply solve -B ≤ A ≤ B. This might involve algebraic manipulation to isolate the variable.

2. A is non-negative and B is negative: If A ≥ 0 and B < 0, then the inequality -B ≤ A ≤ B becomes a contradiction since -B will be positive. In this scenario, there is no solution unless we have A=0 and B=0 which is consistent with |A| ≤ |B|.

3. A is negative and B is non-negative: If A < 0 and B ≥ 0, the inequality -B ≤ A ≤ B might yield a solution. We need to solve for A within the range specified.

4. A and B are both negative: If A < 0 and B < 0, we again solve the compound inequality -B ≤ A ≤ B. The solution set might include negative values.

Case 2: |A| ≥ |B|

This inequality signifies that the absolute value of A is greater than or equal to the absolute value of B. This translates to two separate inequalities:

1. A ≥ B
2. A ≤ -B

We need to solve each inequality individually. The solution set will be the union of the solution sets for both inequalities.

Let's illustrate these principles with several examples.

Example Problems: Solving Inequalities with Absolute Values on Both Sides

Example 1: |2x - 1| ≤ |x + 3|

This inequality follows the form |A| ≤ |B|. We use the compound inequality:

• |x + 3| ≤ 2x - 1 ≤ |x + 3|

We need to consider cases based on the signs of (2x - 1) and (x + 3).

Case 1: 2x - 1 ≥ 0 and x + 3 ≥ 0

This means x ≥ 1/2 and x ≥ -3. Thus, x ≥ 1/2. The inequality becomes:

-(x + 3) ≤ 2x - 1 ≤ x + 3

Solving this compound inequality yields: -2 ≤ x ≤ 4. Since we must also have x ≥ 1/2, the solution for this case is 1/2 ≤ x ≤ 4.

Case 2: 2x - 1 ≥ 0 and x + 3 < 0

This implies x ≥ 1/2 and x < -3, which is a contradiction. Therefore, there is no solution in this case.

Case 3: 2x - 1 < 0 and x + 3 ≥ 0

This means x < 1/2 and x ≥ -3. So, -3 ≤ x < 1/2. The inequality becomes:

-(x + 3) ≤ -(2x - 1) ≤ x + 3

Solving this yields -4 ≤ x ≤ 4. Since we must satisfy -3 ≤ x < 1/2, the solution for this case is -3 ≤ x < 1/2.

Case 4: 2x - 1 < 0 and x + 3 < 0

This is x < 1/2 and x < -3. This means x < -3. The inequality becomes:

-(x + 3) ≤ -(2x - 1) ≤ x + 3

Solving gives -4 ≤ x ≤ 4. Since we have x < -3, the solution for this case is x ≤ -4.

Combining all cases, the final solution is -3 ≤ x ≤ 4.

Example 2: |3x - 2| ≥ |x + 1|

This is a |A| ≥ |B| inequality. We have two separate inequalities to solve:

1. 3x - 2 ≥ x + 1
2. 3x - 2 ≤ -(x + 1)

Solving the first inequality:

3x - 2 ≥ x + 1 => 2x ≥ 3 => x ≥ 3/2

Solving the second inequality:

3x - 2 ≤ -x - 1 => 4x ≤ 1 => x ≤ 1/4

Therefore, the solution is x ≤ 1/4 or x ≥ 3/2.

Advanced Techniques and Considerations

• Graphing: Visualizing the inequalities by graphing the absolute value functions can be insightful, particularly for more complex scenarios. The solution will represent the regions where one function's graph lies above or below the other.

• Squaring both sides: While generally discouraged due to potential introduction of extraneous solutions, squaring both sides can simplify certain inequalities, especially when dealing with squared terms inside the absolute value. However, always check for extraneous solutions afterward.

• Test points: After solving, test points within and outside the solution intervals to verify your solution is correct.

Conclusion: Mastering Absolute Value Inequalities

Solving inequalities with absolute values on both sides demands a careful and systematic approach. By understanding the principles of absolute value and employing the case-by-case analysis outlined in this guide, you can confidently solve even the most challenging inequalities. Remember to check your solutions and use visualization techniques like graphing to enhance your understanding and problem-solving skills. Practice is key to mastering this important mathematical concept. Consistent practice with a variety of examples will build your confidence and proficiency. Remember to always double-check your work and consider using different methods to verify your answers.