Solving Absolute Value Equation |x+4|-5=6 A Step-by-Step Guide

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Absolute value equations can seem daunting at first, but with a systematic approach, they become quite manageable. This article will delve into the process of solving the absolute value equation $|x+4|-5=6$, providing a clear, step-by-step solution and offering insights into the underlying concepts. Understanding absolute value is crucial in various mathematical fields, making it an essential skill for students and professionals alike. This guide aims to not only solve the given equation but also to equip you with the knowledge to tackle similar problems with confidence.

Understanding Absolute Value

Before diving into the solution, it's essential to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line. Distance is always non-negative, so the absolute value of a number is always non-negative. Mathematically, the absolute value of a number x, denoted as |x|, is defined as:

|x| = x, if x ≥ 0 |x| = -x, if x < 0

For example, |5| = 5 because 5 is 5 units away from zero. Similarly, |-5| = 5 because -5 is also 5 units away from zero. This fundamental understanding is key to solving equations involving absolute values. When we encounter an equation like $|x+4|-5=6$, we need to consider both possibilities: the expression inside the absolute value bars is either positive or negative. This leads to two separate equations that need to be solved independently. The absolute value concept is not just a mathematical abstraction; it has practical applications in various fields, including physics, engineering, and computer science, where dealing with magnitudes and distances is crucial. Therefore, mastering this concept is a valuable investment in your mathematical journey.

The Importance of Absolute Value in Mathematics

In mathematics, absolute value plays a pivotal role in various areas, including algebra, calculus, and real analysis. It provides a way to measure the magnitude of a number without considering its sign. This is particularly useful in situations where direction or sign is irrelevant, and only the magnitude matters. For example, when calculating the distance between two points on a number line, we use absolute value to ensure the result is always positive. The absolute value function is also crucial in defining concepts like limits and continuity in calculus. Understanding the properties of absolute value is essential for solving inequalities, graphing functions, and analyzing the behavior of mathematical models. Furthermore, absolute value is used in defining norms in vector spaces, which are fundamental in linear algebra and functional analysis. The versatility of absolute value makes it a fundamental tool in the mathematical arsenal, and a solid grasp of its properties is crucial for success in higher-level mathematics. Therefore, taking the time to fully understand absolute value is an investment that pays off in numerous ways throughout your mathematical studies and beyond.

Solving the Equation $|x+4|-5=6$

Now, let's tackle the equation $|x+4|-5=6$. The first step is to isolate the absolute value expression. We can do this by adding 5 to both sides of the equation:

$|x+4|-5+5=6+5$

$|x+4|=11$

Now that we have the absolute value expression isolated, we can consider the two possible cases:

Case 1: The expression inside the absolute value is positive or zero.

In this case, $x+4$ is either positive or zero, so $|x+4| = x+4$. Thus, we have the equation:

$x+4=11$

Subtracting 4 from both sides, we get:

$x=11-4$

$x=7$

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

In this case, $x+4$ is negative, so $|x+4| = -(x+4)$. Thus, we have the equation:

$-(x+4)=11$

Distributing the negative sign, we get:

$-x-4=11$

Adding 4 to both sides, we get:

$-x=11+4$

$-x=15$

Multiplying both sides by -1, we get:

$x=-15$

Therefore, the solutions to the equation $|x+4|-5=6$ are $x=7$ and $x=-15$.

Detailed Breakdown of Each Step

To further clarify the solution process, let's break down each step in detail. Isolating the absolute value is the crucial first step because it allows us to apply the definition of absolute value effectively. By adding 5 to both sides, we ensure that the absolute value expression is the only term on one side of the equation. This sets the stage for considering the two possible cases. In Case 1, we assume that the expression inside the absolute value, $x+4$, is non-negative. This means that the absolute value is simply the expression itself, allowing us to write $x+4=11$. Solving this linear equation involves subtracting 4 from both sides, which gives us the solution $x=7$. In Case 2, we consider the possibility that $x+4$ is negative. In this scenario, the absolute value is the negation of the expression, leading to the equation $-(x+4)=11$. Distributing the negative sign and simplifying, we get $-x-4=11$. Adding 4 to both sides gives us $-x=15$, and multiplying by -1 yields the solution $x=-15$. By meticulously examining each step, we gain a deeper understanding of the solution process and reinforce the underlying principles of absolute value equations.

Verification of Solutions

It's always a good practice to verify the solutions we obtain. To verify the solutions $x=7$ and $x=-15$, we substitute them back into the original equation $|x+4|-5=6$.

For $x=7$:

$|7+4|-5=6$

$|11|-5=6$

$11-5=6$

$6=6$ (True)

For $x=-15$:

$|-15+4|-5=6$

$|-11|-5=6$

$11-5=6$

$6=6$ (True)

Both solutions satisfy the original equation, so our solutions are correct.

The Importance of Verification in Mathematical Problem Solving

Verifying solutions is an indispensable step in mathematical problem-solving. It serves as a safeguard against errors that may arise during the solution process. By substituting the obtained values back into the original equation, we confirm whether they satisfy the given conditions. This process helps identify mistakes in algebraic manipulations, arithmetic calculations, or logical reasoning. In the context of absolute value equations, verification is particularly crucial because the process of splitting the equation into two cases can sometimes introduce extraneous solutions. These are solutions that emerge from the algebraic steps but do not satisfy the original equation. By verifying each solution, we can eliminate any extraneous values and ensure that our final answers are accurate. Furthermore, the act of verification reinforces our understanding of the problem and the solution process. It allows us to critically evaluate our work and gain confidence in the correctness of our answers. Therefore, cultivating the habit of verifying solutions is a fundamental aspect of mathematical proficiency and enhances the reliability of our results.

Conclusion

In this article, we have successfully solved the absolute value equation $|x+4|-5=6$. We found that the solutions are $x=7$ and $x=-15$. We achieved this by first understanding the concept of absolute value, then isolating the absolute value expression, considering the two possible cases, and finally, verifying our solutions. This step-by-step approach can be applied to solve a wide range of absolute value equations. Remember, the key to mastering absolute value equations is to understand the definition of absolute value and to systematically consider all possible cases. With practice, you can confidently tackle these types of problems.

Further Practice and Resources for Absolute Value Equations

To solidify your understanding of absolute value equations, consistent practice is key. Start by working through a variety of examples, gradually increasing the complexity of the problems. Focus on understanding the underlying principles and applying the step-by-step approach discussed in this article. Online resources such as Khan Academy, Mathway, and Wolfram Alpha offer numerous practice problems and detailed solutions. Textbooks and workbooks also provide a wealth of exercises and examples. Additionally, consider working with a tutor or joining a study group to discuss challenging problems and share insights. Exploring different types of absolute value equations, including those involving inequalities and more complex expressions, will further enhance your skills. Remember to always verify your solutions to ensure accuracy. By dedicating time and effort to practice, you can develop a strong command of absolute value equations and improve your overall mathematical proficiency. This skill will not only help you in academic settings but also in real-world situations where mathematical problem-solving is required.

Final Answer: The final answer is (A) $x=7$ and $x=-15$.