Solve |x-12|-4 < 20 Absolute Value Inequality In Interval Notation

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Solving absolute value inequalities is a fundamental skill in algebra and precalculus. This article provides a comprehensive guide to solving the absolute value inequality $|x-12|-4 < 20$. We will walk through the steps necessary to find the solution set and express it in interval notation. Interval notation is a standard way to represent a set of real numbers, using intervals defined by endpoints and parentheses or brackets to indicate whether the endpoints are included. Understanding how to manipulate absolute value inequalities is crucial for various mathematical applications, including calculus, optimization problems, and real-world scenarios involving constraints and tolerances. The absolute value of a number represents its distance from zero on the number line. When dealing with inequalities involving absolute values, we are essentially looking for the range of values that satisfy a given distance condition. This often results in compound inequalities, which can be solved by breaking them down into simpler inequalities and considering the different cases arising from the absolute value definition. In the given inequality, $|x-12|-4 < 20$, the absolute value term $|x-12|$ represents the distance between $x$ and $12$. The inequality states that this distance, after subtracting 4, must be less than 20. To solve this, we need to isolate the absolute value term, consider both positive and negative cases, and express the final solution in interval notation. This process involves algebraic manipulation, careful attention to detail, and a solid understanding of interval notation conventions. By mastering the techniques for solving absolute value inequalities, you will enhance your problem-solving skills and gain a deeper appreciation for the elegance and precision of mathematical notation.

Understanding Absolute Value Inequalities

Before diving into the specific problem, let's discuss the general principles behind absolute value inequalities. The absolute value of a number $a$, denoted as $|a|$, is its distance from 0 on the number line. Mathematically, this is defined as:

|a| = egin{cases} a, & ext{if } a ext{ is greater than or equal to } 0 \ -a, & ext{if } a < 0 ext{ (if a is less than 0)} ext{ less than } 0 \end{cases}

When solving an absolute value inequality, we must consider two cases due to this definition. For an inequality of the form $|x| < a$, where $a$ is a positive number, we have:

$$-a < x < a$$

This means $x$ must be within the interval $(-a, a)$. Similarly, for an inequality of the form $|x| > a$, we have two separate cases:

$$x < -a ext{ or } x > a$$

This indicates that $x$ must be either less than $-a$ or greater than $a$. These principles form the foundation for solving more complex absolute value inequalities. Understanding the geometric interpretation of absolute value as distance is essential for grasping these concepts. When we say $|x| < a$, we are essentially saying that the distance of $x$ from 0 is less than $a$. This corresponds to all points between $-a$ and $a$ on the number line. On the other hand, $|x| > a$ means the distance of $x$ from 0 is greater than $a$, which includes all points to the left of $-a$ and to the right of $a$. These visualizations help in understanding why we split absolute value inequalities into two cases. The first case deals with the positive scenario, where the expression inside the absolute value is already non-negative, and the absolute value signs can be removed directly. The second case addresses the negative scenario, where the expression inside the absolute value is negative, and we need to negate the expression before removing the absolute value signs. By carefully considering both cases, we ensure that we capture all possible solutions to the inequality. Furthermore, it is crucial to remember that the solution to an absolute value inequality is often a set of values rather than a single value. This set can be represented in various ways, including set-builder notation, interval notation, and graphically on the number line. The choice of representation depends on the specific context and the desired level of clarity. In this article, we will focus on expressing the solution in interval notation, which is a concise and widely used method for representing intervals of real numbers.

Step-by-Step Solution

Now, let's solve the given inequality step-by-step:

$$|x-12|-4 < 20$$

Step 1: Isolate the Absolute Value

First, we need to isolate the absolute value term. To do this, add 4 to both sides of the inequality:

$$|x-12|-4 + 4 < 20 + 4$$

$$|x-12| < 24$$

Step 2: Set Up Two Cases

Now we have the absolute value isolated, we can set up two cases based on the definition of absolute value.

Case 1: The expression inside the absolute value is non-negative:

$$x-12 < 24$$

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

$$-(x-12) < 24$$

Step 3: Solve Each Case

Let's solve each case separately.

Case 1:

$$x-12 < 24$$

Add 12 to both sides:

$$x < 36$$

Case 2:

$$-(x-12) < 24$$

Distribute the negative sign:

$$-x+12 < 24$$

Subtract 12 from both sides:

$$-x < 12$$

Multiply both sides by -1 (and remember to flip the inequality sign because we're multiplying by a negative number):

$$x > -12$$

Step 4: Combine the Solutions

We have two inequalities:

$$x < 36 ext{ and } x > -12$$

These inequalities can be combined into a single compound inequality:

$$-12 < x < 36$$

Step 5: Express the Solution in Interval Notation

Finally, we express the solution in interval notation. The inequality $-12 < x < 36$ means that $x$ can be any number between -12 and 36, not including the endpoints. Therefore, the interval notation is:

$$(-12, 36)$$

This interval represents all real numbers greater than -12 and less than 36. In interval notation, parentheses indicate that the endpoints are not included, while brackets would indicate that the endpoints are included. For example, if the inequality had been $-12 ext{≤} x ext{≤} 36$, the interval notation would be $[-12, 36]$. It is essential to use the correct notation to accurately represent the solution set. The interval notation provides a concise and clear way to communicate the range of values that satisfy the given inequality. Understanding how to convert inequalities into interval notation and vice versa is a crucial skill in mathematics, particularly in calculus and analysis, where intervals are frequently used to define domains, ranges, and other important concepts.

Common Mistakes and How to Avoid Them

Solving absolute value inequalities can be tricky, and several common mistakes can lead to incorrect answers. Being aware of these pitfalls and how to avoid them is essential for mastering the process. Here are some of the most frequent errors:

1. Forgetting to consider both cases: The most common mistake is failing to split the absolute value inequality into two separate cases. Remember that the absolute value of an expression can be either the expression itself or its negation. Therefore, you must consider both scenarios to find all possible solutions. For example, if you are solving $|x-5| < 3$, you must consider both $x-5 < 3$ and $-(x-5) < 3$. Ignoring one of these cases will lead to an incomplete solution.

2. Flipping the inequality sign incorrectly: When multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality sign. This is a crucial rule that is often overlooked. For example, when solving $-x < 4$, you must multiply both sides by -1 and flip the inequality sign to get $x > -4$. Failing to do so will result in an incorrect solution set.

3. Incorrectly distributing the negative sign: When dealing with the negative case of an absolute value inequality, such as $-(x-3) < 2$, it is essential to distribute the negative sign correctly. The expression becomes $-x + 3 < 2$, not $-x - 3 < 2$. A simple mistake in distributing the sign can lead to a completely different solution.

4. Mixing up "and" and "or" conditions: When combining the solutions from the two cases, it is crucial to determine whether the solutions should be combined with an "and" or an "or" condition. For inequalities of the form $|x| < a$, the solutions are combined with "and," indicating an intersection of intervals. For inequalities of the form $|x| > a$, the solutions are combined with "or," indicating a union of intervals. Confusing these conditions will lead to an incorrect representation of the solution set.

5. Misinterpreting interval notation: Interval notation is a concise way to represent sets of numbers, but it can be confusing if not understood properly. Parentheses ( ) indicate that the endpoint is not included, while brackets [ ] indicate that the endpoint is included. For example, $(2, 5)$ represents all numbers between 2 and 5, excluding 2 and 5, while $[2, 5]$ includes both 2 and 5. Pay close attention to these symbols when expressing the final solution.

6. Not isolating the absolute value first: Before splitting the inequality into cases, you must isolate the absolute value term. For example, in the inequality $2|x+1| - 3 < 5$, you should first add 3 to both sides and then divide by 2 to get $|x+1| < 4$. Solving the inequality without isolating the absolute value term will lead to incorrect results.

By being mindful of these common mistakes and practicing the steps involved in solving absolute value inequalities, you can significantly improve your accuracy and understanding. Always double-check your work and ensure that your solution makes sense in the context of the original problem. If possible, graph the solution on a number line to visually confirm that your answer is correct.

Conclusion

In this article, we have demonstrated how to solve the absolute value inequality $|x-12|-4 < 20$. By isolating the absolute value, considering both positive and negative cases, solving the resulting inequalities, and expressing the solution in interval notation, we found the solution to be $(-12, 36)$. Solving absolute value inequalities is a crucial skill in algebra, with applications in various mathematical fields. The key to success lies in understanding the definition of absolute value, breaking the problem into cases, and meticulously applying algebraic techniques. This skill not only enhances your mathematical proficiency but also builds a solid foundation for more advanced topics in mathematics and related disciplines. The ability to solve absolute value inequalities is essential for tackling problems in calculus, optimization, and real-world applications involving constraints and tolerances. Whether you are a student preparing for an exam or a professional working on a complex problem, mastering this skill will undoubtedly prove beneficial. Furthermore, the process of solving absolute value inequalities reinforces critical thinking and problem-solving skills. It teaches you to break down complex problems into smaller, more manageable parts and to consider all possible scenarios. This methodical approach is valuable not only in mathematics but also in various aspects of life. By understanding the underlying principles and practicing regularly, you can confidently approach absolute value inequalities and other mathematical challenges.