Solving |3m+4|+6=12 A Step By Step Guide

by Jeany 41 views
Iklan Headers

Introduction: Understanding Absolute Value Equations

In mathematics, absolute value represents the distance of a number from zero on the number line. This distance is always non-negative. The absolute value of a number x is denoted as |x|. Absolute value equations are equations that involve an absolute value expression. Solving these equations requires a careful approach to account for the two possible cases arising from the absolute value: the expression inside the absolute value can be either positive or negative. In this article, we will delve into the process of solving the absolute value equation |3m+4|+6=12, providing a comprehensive, step-by-step explanation to enhance your understanding and problem-solving skills.

Before we dive into the specifics of our equation, let's solidify our understanding of absolute value. The absolute value of a number is its magnitude, irrespective of its sign. For instance, |5| = 5 and |-5| = 5. This is because both 5 and -5 are 5 units away from zero. When solving equations involving absolute values, we must consider both the positive and negative possibilities of the expression inside the absolute value bars. This is the core concept that governs the methodology for solving absolute value equations.

The process of solving absolute value equations often involves isolating the absolute value expression first. This means performing algebraic manipulations to get the absolute value term alone on one side of the equation. Once isolated, we can then split the equation into two separate cases. The first case considers the expression inside the absolute value as positive or zero, while the second case considers it as negative. Each case leads to a different linear equation, which we can solve using standard algebraic techniques. The solutions obtained from both cases together form the complete solution set for the original absolute value equation.

Absolute value equations appear frequently in various mathematical contexts, including algebra, calculus, and real analysis. They also have practical applications in fields like physics, engineering, and computer science, where dealing with magnitudes and distances is crucial. Mastering the techniques for solving these equations is therefore essential for a solid foundation in mathematical problem-solving. In the following sections, we will meticulously dissect the equation |3m+4|+6=12, illustrating each step with clarity and precision, ensuring you grasp the underlying principles involved.

Step 1: Isolate the Absolute Value Expression

To effectively solve the absolute value equation |3m+4|+6=12, the initial and crucial step is to isolate the absolute value expression. This involves manipulating the equation algebraically to get the term |3m+4| alone on one side. The presence of the '+6' outside the absolute value bars necessitates its removal to achieve this isolation. To do so, we apply the inverse operation, which in this case is subtraction. We subtract 6 from both sides of the equation, maintaining the balance and equality of the expression. This step is fundamental because it sets the stage for addressing the two possible scenarios that arise from the absolute value: the expression inside the absolute value bars can be either positive or negative.

The original equation is |3m+4|+6=12. Subtracting 6 from both sides, we get |3m+4|+6-6=12-6. This simplifies to |3m+4|=6. By performing this subtraction, we have successfully isolated the absolute value expression |3m+4| on the left side of the equation. This isolation is critical because it allows us to directly address the core concept of absolute value, which is the distance from zero. With the absolute value expression isolated, we can now proceed to the next step, which involves considering the two cases that stem from the definition of absolute value.

The importance of isolating the absolute value expression cannot be overstated. It is analogous to simplifying an equation before applying other solution techniques. Just as we simplify algebraic equations by combining like terms or distributing factors, isolating the absolute value allows us to focus solely on the absolute value component of the equation. This clarity is essential for accurate problem-solving. Without this isolation, we would be dealing with a more complex expression, making it difficult to apply the fundamental principles of absolute value. In essence, this step prepares the equation for the application of the absolute value definition, leading us closer to the solution.

Once we have isolated the absolute value, we can move forward with confidence, knowing that the subsequent steps will build upon this solid foundation. The equation |3m+4|=6 now clearly presents us with the core challenge: determining the values of m that make the expression 3m+4 have a distance of 6 from zero. This understanding is the key to unlocking the solution. In the subsequent sections, we will explore the two cases that arise from this understanding, leading us to the complete solution set for the equation.

Step 2: Setting Up Two Cases

After successfully isolating the absolute value expression in the equation |3m+4|=6, the next crucial step is to set up two distinct cases. This is because the absolute value of a quantity can be equal to a specific value in two scenarios: when the quantity itself is equal to that value, and when the quantity is equal to the negative of that value. This dual possibility stems directly from the definition of absolute value as the distance from zero, which is always non-negative. Therefore, to solve the equation completely, we must consider both scenarios separately.

Case 1 arises when the expression inside the absolute value, 3m+4, is equal to the positive value on the other side of the equation, which is 6. This case represents the scenario where 3m+4 is already a positive number or zero, and its distance from zero is 6. Mathematically, we express this case as 3m+4=6. This equation is a linear equation that can be solved using standard algebraic techniques, such as subtracting 4 from both sides and then dividing by 3. The solution obtained from this case will be one part of the complete solution set for the original absolute value equation.

Case 2, on the other hand, arises when the expression inside the absolute value, 3m+4, is equal to the negative of the value on the other side of the equation, which is -6. This case represents the scenario where 3m+4 is a negative number, and its absolute value (distance from zero) is 6. Mathematically, we express this case as 3m+4=-6. Similar to Case 1, this equation is also a linear equation and can be solved using the same algebraic techniques. The solution obtained from this case will be the second part of the complete solution set. It's important to recognize that both cases are equally valid and must be considered to find all possible solutions.

By setting up these two cases, we are essentially breaking down the absolute value equation into two simpler linear equations. This strategy is fundamental to solving absolute value equations because it directly addresses the dual nature of absolute value. Failing to consider both cases would lead to an incomplete solution set, missing one or more possible values for m. The solutions from each case will represent the values of m that satisfy the original absolute value equation, and together, they form the solution set. In the following sections, we will solve each of these cases independently, illustrating the algebraic steps involved and highlighting the importance of each step in arriving at the correct solutions.

Step 3: Solving Case 1 (3m+4=6)

Having established Case 1 as 3m+4=6, we now proceed to solve this linear equation for m. The goal is to isolate the variable m on one side of the equation, revealing its value. This involves a series of algebraic manipulations, carefully applied to maintain the equation's balance. The first step in isolating m is to remove the constant term, +4, from the left side of the equation. We achieve this by performing the inverse operation, which is subtraction. Subtracting 4 from both sides of the equation preserves the equality and moves us closer to isolating m.

Subtracting 4 from both sides of 3m+4=6, we get 3m+4-4=6-4. This simplifies to 3m=2. Now, we have a simpler equation where the term with m is the only term on the left side. To completely isolate m, we need to remove the coefficient 3 that is multiplying it. We again use the inverse operation, which in this case is division. Dividing both sides of the equation by 3 will isolate m and give us its value.

Dividing both sides of 3m=2 by 3, we get (3m)/3=2/3. This simplifies to m=2/3. Therefore, the solution for m in Case 1 is 2/3. This value represents one of the possible solutions to the original absolute value equation. It is crucial to remember that this is only one part of the solution set, as we still need to consider Case 2, which arises from the negative possibility of the absolute value expression. However, by methodically solving Case 1, we have demonstrated the algebraic techniques involved in isolating the variable and finding a solution.

The process of solving linear equations like this is a fundamental skill in algebra. It involves applying inverse operations in a strategic manner to isolate the variable of interest. Subtraction and division are key tools in this process, and understanding when and how to apply them is essential for success in solving equations. In this case, subtracting 4 and then dividing by 3 were the necessary steps to reveal the value of m. This methodical approach not only gives us the solution but also builds confidence in our ability to handle similar algebraic problems. In the next section, we will turn our attention to Case 2, where we will apply the same principles to find the other possible solution for m.

Step 4: Solving Case 2 (3m+4=-6)

Having determined the solution for Case 1, we now shift our focus to Case 2, which is represented by the equation 3m+4=-6. This case arises from the possibility that the expression inside the absolute value, 3m+4, could be negative, with its absolute value being equal to 6. The process of solving this equation mirrors that of Case 1, involving algebraic manipulations to isolate the variable m. Our primary goal remains the same: to find the value of m that satisfies this equation.

As in Case 1, the first step towards isolating m is to remove the constant term, +4, from the left side of the equation. We accomplish this by applying the inverse operation, which is subtraction. Subtracting 4 from both sides of the equation ensures that the equality is maintained while moving us closer to isolating m. This step is crucial because it simplifies the equation, allowing us to focus on the term containing m.

Subtracting 4 from both sides of 3m+4=-6, we get 3m+4-4=-6-4. This simplifies to 3m=-10. Now, we have a simpler equation where the only term on the left side is 3m. To completely isolate m, we need to eliminate the coefficient 3 that is multiplying it. We once again employ the inverse operation, which is division. Dividing both sides of the equation by 3 will isolate m and reveal its value.

Dividing both sides of 3m=-10 by 3, we get (3m)/3=-10/3. This simplifies to m=-10/3. Therefore, the solution for m in Case 2 is -10/3. This value represents the second possible solution to the original absolute value equation. It is important to note that this solution is distinct from the solution obtained in Case 1, highlighting the necessity of considering both cases when solving absolute value equations. By systematically solving Case 2, we have completed the process of finding all possible solutions for m.

The algebraic techniques used in solving Case 2 are identical to those used in Case 1, demonstrating the consistency of the problem-solving approach. Subtracting a constant and then dividing by a coefficient are common strategies in solving linear equations. The key is to apply these operations carefully and maintain the balance of the equation at each step. With the solution for Case 2 in hand, we now have two potential values for m that satisfy the original absolute value equation. In the next section, we will combine these solutions to form the complete solution set and verify their validity.

Step 5: Forming the Solution Set and Verification

Having solved both Case 1 and Case 2, we have obtained two potential solutions for the absolute value equation |3m+4|+6=12. In Case 1, we found m=2/3, and in Case 2, we found m=-10/3. The solution set for the equation is the collection of all values of m that satisfy the equation. Therefore, we combine the solutions from both cases to form the solution set. However, before we definitively declare this as the final answer, it is crucial to verify these solutions. Verification ensures that the values we have found indeed satisfy the original equation and that no errors were made during the solving process.

The solution set is represented as {2/3, -10/3}. To verify these solutions, we substitute each value of m back into the original equation |3m+4|+6=12 and check if the equation holds true. This process is essential to confirm the accuracy of our solutions and to guard against any extraneous solutions that might have arisen due to the absolute value nature of the equation.

First, let's verify m=2/3. Substituting this value into the original equation, we get |3*(2/3)+4|+6=12. Simplifying the expression inside the absolute value, we have |2+4|+6=12, which becomes |6|+6=12. Since the absolute value of 6 is 6, the equation simplifies further to 6+6=12, which is indeed true. This confirms that m=2/3 is a valid solution.

Next, let's verify m=-10/3. Substituting this value into the original equation, we get |3*(-10/3)+4|+6=12. Simplifying the expression inside the absolute value, we have |-10+4|+6=12, which becomes |-6|+6=12. The absolute value of -6 is 6, so the equation simplifies to 6+6=12, which is also true. This confirms that m=-10/3 is a valid solution as well.

Since both values of m satisfy the original equation, we can confidently state that the solution set for the equation |3m+4|+6=12 is {2/3, -10/3}. This set contains all the values of m that make the equation true, and we have verified their validity through substitution. The verification step is a critical component of the problem-solving process, ensuring the accuracy and completeness of the solution.

Conclusion: Summarizing the Solution to |3m+4|+6=12

In conclusion, we have successfully navigated the process of solving the absolute value equation |3m+4|+6=12, arriving at the complete solution set. The journey involved several key steps, each building upon the previous one to methodically unravel the equation's complexities. We began by understanding the fundamental concept of absolute value, which is the distance of a number from zero, and how this concept necessitates considering both positive and negative possibilities when solving equations.

The first step in our solution process was to isolate the absolute value expression, |3m+4|, on one side of the equation. This was achieved by subtracting 6 from both sides, resulting in the simplified equation |3m+4|=6. This isolation was crucial because it allowed us to directly address the absolute value component of the equation without any extraneous terms interfering.

Following the isolation, we set up two distinct cases to account for the two possibilities arising from the absolute value. Case 1 considered the scenario where the expression inside the absolute value, 3m+4, is equal to the positive value on the other side of the equation, which is 6. This led to the linear equation 3m+4=6. Case 2, on the other hand, considered the scenario where 3m+4 is equal to the negative of the value, which is -6, resulting in the linear equation 3m+4=-6.

We then proceeded to solve each case independently using standard algebraic techniques. In Case 1, we subtracted 4 from both sides and then divided by 3, yielding the solution m=2/3. Similarly, in Case 2, we subtracted 4 from both sides and then divided by 3, yielding the solution m=-10/3. These two solutions represent the potential values of m that satisfy the original equation.

To ensure the validity of our solutions, we performed a crucial verification step. We substituted each value of m back into the original equation and confirmed that the equation held true in both instances. This step is vital in preventing extraneous solutions and ensuring the accuracy of the final answer.

Finally, we combined the validated solutions to form the solution set, which is {2/3, -10/3}. This set represents all the values of m that satisfy the equation |3m+4|+6=12. The successful completion of this problem demonstrates the importance of a systematic approach to solving absolute value equations, encompassing isolation, case consideration, algebraic manipulation, and verification. The principles and techniques applied here can be extended to solve a wide range of absolute value equations, enhancing your problem-solving capabilities in mathematics.