Solving Absolute Value Equations A Step-by-Step Guide For M = -4|m + 8| + 10 = 6
Introduction
In this article, we will delve into the process of solving the equation m = -4|m + 8| + 10 = 6. This equation involves an absolute value, which adds a layer of complexity. We will break down the steps systematically, ensuring a clear and thorough understanding of how to arrive at the solution. Equations involving absolute values often require careful consideration of different cases, making it crucial to approach the problem with a methodical strategy. By the end of this guide, you will not only be able to solve this particular equation but also gain valuable insights into tackling similar mathematical challenges. Solving equations is a fundamental skill in mathematics, and mastering techniques for equations involving absolute values can significantly enhance your problem-solving capabilities. This guide will walk you through each step, providing clear explanations and justifications to ensure you grasp the underlying concepts fully. Understanding how to handle absolute values in equations is essential for various fields, including engineering, physics, and computer science, where such problems frequently arise. So, let's embark on this journey to unravel the solution to m = -4|m + 8| + 10 = 6. Our exploration will cover the definition of absolute value, the different scenarios that arise when dealing with it in equations, and the algebraic manipulations needed to isolate the variable m. We'll also discuss how to check our solutions to ensure they are valid and do not introduce extraneous roots. This comprehensive approach will equip you with the skills necessary to confidently solve equations with absolute values.
Understanding Absolute Value
Before diving into the specifics of solving the equation, it is crucial to have a solid grasp of the concept of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. Mathematically, the absolute value of a real number x, denoted as |x|, is defined as follows:
|x| = x, if x ≥ 0 |x| = -x, if x < 0
This definition implies that the absolute value of a positive number or zero is the number itself, while the absolute value of a negative number is its negation (making it positive). For example, |5| = 5, and |-5| = -(-5) = 5. Understanding this fundamental concept is essential when dealing with equations that involve absolute values. When an equation contains an absolute value expression, such as |m + 8|, it means that the expression inside the absolute value bars, m + 8 in this case, can be either positive or negative. This gives rise to two separate cases that we need to consider to find all possible solutions for m. In our equation, m = -4|m + 8| + 10 = 6, the presence of the absolute value term |m + 8| necessitates that we analyze two distinct scenarios: one where m + 8 is non-negative (i.e., m + 8 ≥ 0) and another where m + 8 is negative (i.e., m + 8 < 0). By examining these two cases separately, we can remove the absolute value and solve the resulting equations to find potential values for m. However, it's crucial to remember that not all solutions obtained from these cases may be valid for the original equation due to the nature of the absolute value. Therefore, we must check our solutions against the initial conditions imposed by the absolute value definition. This process ensures that we only accept solutions that satisfy the original equation and its constraints.
Step-by-Step Solution
Now, let’s proceed with solving the equation m = -4|m + 8| + 10 = 6. The first step is to isolate the absolute value term. Subtract 10 from both sides of the equation:
m = -4|m + 8| + 10 = 6
Subtracting 10 from both sides:
m - 10 = -4|m + 8| + 10 - 10
m - 10 = -4|m + 8|
Next, divide both sides by -4:
(m - 10) / -4 = |m + 8|
This simplifies to:
|m + 8| = (10 - m) / 4
Now, we consider two cases based on the definition of absolute value:
Case 1: m + 8 ≥ 0 (or m ≥ -8)
In this case, |m + 8| = m + 8. So, the equation becomes:
m + 8 = (10 - m) / 4
Multiply both sides by 4 to eliminate the fraction:
4(m + 8) = 10 - m
4m + 32 = 10 - m
Add m to both sides:
5m + 32 = 10
Subtract 32 from both sides:
5m = -22
Divide by 5:
m = -22/5
Since -22/5 = -4.4, which is greater than -8, this solution is valid for this case.
Case 2: m + 8 < 0 (or m < -8)
In this case, |m + 8| = -( m + 8). So, the equation becomes:
-(m + 8) = (10 - m) / 4
Multiply both sides by 4:
-4(m + 8) = 10 - m
-4m - 32 = 10 - m
Add 4m to both sides:
-32 = 10 + 3m
Subtract 10 from both sides:
-42 = 3m
Divide by 3:
m = -14
Since -14 is less than -8, this solution is also valid for this case.
Verification of Solutions
Having obtained two potential solutions for solving the equation, it's crucial to verify them in the original equation to ensure they are valid and not extraneous roots. We will substitute each solution back into the equation m = -4|m + 8| + 10 = 6 and check if it holds true.
Verification of m = -22/5
Substitute m = -22/5 into the original equation:
-22/5 = -4|-22/5 + 8| + 10 = 6
First, simplify the expression inside the absolute value:
-22/5 + 8 = -22/5 + 40/5 = 18/5
So, the equation becomes:
-22/5 = -4|18/5| + 10 = 6
-22/5 = -4(18/5) + 10 = 6
-22/5 = -72/5 + 50/5 = 6
-22/5 = -22/5 = 6
The left side equals the right side, confirming that m = -22/5 is a valid solution.
Verification of m = -14
Substitute m = -14 into the original equation:
-14 = -4|-14 + 8| + 10 = 6
Simplify the expression inside the absolute value:
-14 + 8 = -6
So, the equation becomes:
-14 = -4|-6| + 10 = 6
-14 = -4(6) + 10 = 6
-14 = -24 + 10 = 6
-14 = -14 = 6
The left side equals the right side, confirming that m = -14 is a valid solution.
Conclusion
In conclusion, by meticulously following the steps to solve the equation m = -4|m + 8| + 10 = 6, we have identified two valid solutions: m = -22/5 and m = -14. These solutions were obtained by considering the two cases arising from the absolute value expression and subsequently verified in the original equation. This process underscores the importance of understanding the definition of absolute value and applying it carefully when solving equations. Equations involving absolute values often require a case-by-case analysis to ensure that all possible solutions are captured. Ignoring this aspect can lead to incomplete or incorrect solutions. Furthermore, the verification step is crucial in preventing the inclusion of extraneous roots, which may arise due to the algebraic manipulations involved in solving the equations. The techniques employed in this article are applicable to a wide range of equations involving absolute values, making this a valuable skill for anyone studying mathematics or related fields. Mastering these techniques not only enhances your problem-solving abilities but also deepens your understanding of fundamental mathematical concepts. Remember, the key to successfully solving equations with absolute values lies in systematically addressing the different cases and meticulously verifying the solutions. With practice and a clear understanding of the underlying principles, you can confidently tackle even the most challenging problems in this area.
