How To Solve Inequalities With Absolute Value

Absolute value inequalities might seem intimidating at first, but they are actually quite manageable once you understand the underlying principles and follow a systematic approach. This guide provides a comprehensive overview of how to solve inequalities involving absolute values, complete with examples and explanations to help you master this concept.

Understanding Absolute Value

Before diving into inequalities, it's crucial to understand what absolute value represents. The absolute value of a number x, denoted as |x|, is its distance from zero on the number line. This means |x| is always non-negative. For example, |3| = 3 and |-3| = 3.

Definition of Absolute Value

Mathematically, the absolute value of x is defined as:

| x | =

• x, if x ≥ 0
• -x, if x < 0

This definition is the key to solving absolute value inequalities because it tells us that we need to consider two cases: when the expression inside the absolute value is positive or zero, and when it's negative.

Basic Principles of Solving Absolute Value Inequalities

The way you solve an absolute value inequality depends on whether the absolute value expression is less than or greater than a constant. Here's a breakdown of the two primary scenarios:

1. | x | < a: This inequality means that x is within a distance of a from zero. In other words, x must be between -a and a. Therefore, the solution is -a < x < a.

2. | x | > a: This inequality means that x is farther than a distance of a from zero. In other words, x must be less than -a or greater than a. Therefore, the solution is x < -a or x > a.

These principles extend to inequalities with ≤ (less than or equal to) and ≥ (greater than or equal to) symbols as well.

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

Here's a detailed, step-by-step guide to solving absolute value inequalities:

Step 1: Isolate the Absolute Value Expression

The first step is always to isolate the absolute value expression on one side of the inequality. This means performing algebraic operations to get the absolute value term by itself.

Example:

3 | 2x - 1 | + 5 < 14

To isolate the absolute value, we first subtract 5 from both sides:

3 | 2x - 1 | < 9

Then, we divide both sides by 3:

| 2x - 1 | < 3

Now the absolute value expression is isolated.

Step 2: Set up the Two Cases

Based on whether the inequality is of the form | x | < a or | x | > a, set up the two cases that need to be solved.

• For | x | < a, set up -a < x < a
• For | x | > a, set up x < -a or x > a

Example (continuing from above):

Since we have | 2x - 1 | < 3, we set up the following compound inequality:

-3 < 2x - 1 < 3

Step 3: Solve Each Case

Solve each of the inequalities you set up in Step 2. Remember to perform the same operations on all parts of the inequality to maintain balance.

Example (continuing from above):

To solve -3 < 2x - 1 < 3, we first add 1 to all parts of the inequality:

-3 + 1 < 2x - 1 + 1 < 3 + 1

-2 < 2x < 4

Then, we divide all parts by 2:

-2 / 2 < 2x / 2 < 4 / 2

-1 < x < 2

Step 4: Express the Solution

Express the solution in interval notation or set notation. If you have an "or" situation (from | x | > a), you'll have two separate intervals or sets that need to be combined.

Example (continuing from above):

The solution to -1 < x < 2 is the interval (-1, 2). This means all values of x between -1 and 2 (but not including -1 and 2) satisfy the original inequality.

Examples of Solving Absolute Value Inequalities

Let's work through several examples to illustrate the process of solving absolute value inequalities.

Example 1: | x + 3 | ≤ 5

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

2. Set up the two cases: Since we have "≤", we set up:

   -5 ≤ x + 3 ≤ 5

3. Solve each case: Subtract 3 from all parts:

   -5 - 3 ≤ x + 3 - 3 ≤ 5 - 3

   -8 ≤ x ≤ 2

4. Express the solution: The solution is the interval [-8, 2].

Example 2: | 3x - 2 | > 4

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

2. Set up the two cases: Since we have ">", we set up:

   3x - 2 < -4 or 3x - 2 > 4

3. Solve each case:

   • For 3x - 2 < -4:

     Add 2 to both sides: 3x < -2

     Divide by 3: x < -2/3

   • For 3x - 2 > 4:

     Add 2 to both sides: 3x > 6

     Divide by 3: x > 2

4. Express the solution: The solution is x < -2/3 or x > 2. In interval notation, this is (-∞, -2/3) ∪ (2, ∞).

Example 3: 2 | x - 1 | + 3 ≥ 7

1. Isolate the absolute value:

   Subtract 3 from both sides: 2 | x - 1 | ≥ 4

   Divide by 2: | x - 1 | ≥ 2

2. Set up the two cases: Since we have "≥", we set up:

   x - 1 ≤ -2 or x - 1 ≥ 2

3. Solve each case:

   • For x - 1 ≤ -2:

     Add 1 to both sides: x ≤ -1

   • For x - 1 ≥ 2:

     Add 1 to both sides: x ≥ 3

4. Express the solution: The solution is x ≤ -1 or x ≥ 3. In interval notation, this is (-∞, -1] ∪ [3, ∞).

Example 4: | 4 - x | < 3

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

2. Set up the two cases: Since we have "<", we set up:

   -3 < 4 - x < 3

3. Solve each case:

   Subtract 4 from all parts: -7 < -x < -1

   Multiply all parts by -1 (and reverse the inequality signs): 7 > x > 1

   Rewrite: 1 < x < 7

4. Express the solution: The solution is the interval (1, 7).

Special Cases and Considerations

While the above steps cover most absolute value inequalities, there are a few special cases to be aware of:

• Absolute value is always non-negative: If you encounter an inequality like | x | < -2, there is no solution because the absolute value can never be negative. Similarly, if you encounter | x | > -2, the solution is all real numbers because the absolute value is always greater than a negative number.

• Absolute value equals zero: If you have an equation or inequality like | x | = 0, the only solution is x = 0.

• No solution: Sometimes, after isolating the absolute value, you might find an impossible situation. For example, | x + 5 | < -1 has no solution.

• All real numbers: Conversely, an inequality like | x - 2 | ≥ -3 is true for all real numbers.

Solving Absolute Value Inequalities with More Complex Expressions

The same principles apply when the expressions inside the absolute value are more complex, such as quadratic or rational expressions. The key is to carefully follow the steps outlined above and to solve each case correctly.

Example: | x² - 5 | < 4

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

2. Set up the two cases:

   -4 < x² - 5 < 4

3. Solve each case:

   Add 5 to all parts: 1 < x² < 9

   Now we have two separate inequalities to solve:

   • x² > 1: This means x < -1 or x > 1

   • x² < 9: This means -3 < x < 3

4. Combine the solutions: We need to find the values of x that satisfy both conditions.

   x must be greater than 1 but less than 3, or x must be greater than -3 but less than -1.

5. Express the solution: The solution is (-3, -1) ∪ (1, 3).

Example: | (x + 1) / (x - 2) | > 2

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

2. Set up the two cases:

   (x + 1) / (x - 2) < -2 or (x + 1) / (x - 2) > 2

3. Solve each case:

   • For (x + 1) / (x - 2) < -2:

     Add 2 to both sides: (x + 1) / (x - 2) + 2 < 0

     Find a common denominator: (x + 1 + 2(x - 2)) / (x - 2) < 0

     Simplify: (3x - 3) / (x - 2) < 0

     Find critical points: x = 1 and x = 2

     Test intervals: (-∞, 1), (1, 2), (2, ∞)

     The solution is (1, 2)

   • For (x + 1) / (x - 2) > 2:

     Subtract 2 from both sides: (x + 1) / (x - 2) - 2 > 0

     Find a common denominator: (x + 1 - 2(x - 2)) / (x - 2) > 0

     Simplify: (-x + 5) / (x - 2) > 0

     Find critical points: x = 5 and x = 2

     Test intervals: (-∞, 2), (2, 5), (5, ∞)

     The solution is (2, 5)

4. Express the solution: The solution is (1, 2) ∪ (2, 5). Note that x cannot be 2 because it would make the denominator zero.

Practical Applications of Absolute Value Inequalities

Absolute value inequalities are not just abstract mathematical concepts. They have practical applications in various fields:

• Engineering: In engineering, tolerances are often expressed using absolute value inequalities. For example, if a component's length is required to be 10 cm with a tolerance of 0.1 cm, this can be expressed as | L - 10 | ≤ 0.1, where L is the actual length of the component.

• Statistics: In statistics, absolute value inequalities are used to define confidence intervals and to measure the deviation of data points from the mean.

• Economics: In economics, absolute value inequalities can be used to model price fluctuations and to analyze the stability of markets.

• Computer Science: In computer science, they are used in algorithms for error correction and data compression.

Common Mistakes to Avoid

• Forgetting to isolate the absolute value: Always isolate the absolute value expression before setting up the cases.

• Incorrectly setting up the cases: Make sure you understand whether to use "and" or "or" based on the inequality symbol.

• Not reversing the inequality sign when multiplying or dividing by a negative number: Remember to reverse the inequality sign when multiplying or dividing by a negative number.

• Forgetting to check for extraneous solutions: When solving inequalities involving rational expressions, be sure to check for values that make the denominator zero.

• Misinterpreting the solution: Make sure you understand what the solution represents and how to express it correctly in interval notation or set notation.

Conclusion

Solving absolute value inequalities requires a solid understanding of absolute value and a systematic approach. By following the steps outlined in this guide, you can confidently solve a wide range of absolute value inequalities. Remember to practice regularly and to pay attention to the special cases and common mistakes to avoid. With practice, you'll become proficient in solving these types of problems and will be able to apply this knowledge to various real-world scenarios.