Solving Absolute Value Equations Finding All Solutions To Savanah's Math Problem

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In the realm of mathematics, absolute value equations often present a unique challenge, requiring careful consideration of both positive and negative possibilities. In this article, we delve into a specific equation solved by Savanah, exploring her method and uncovering the missing solution. This detailed exploration aims to provide a comprehensive understanding of absolute value equations and their solution strategies. Understanding absolute value equations is crucial for students and math enthusiasts alike. This article breaks down the process step-by-step, ensuring clarity and comprehension. Our journey begins with Savanah's equation: 3+43+4{\frac{x}{2}+3}=11=11. Her initial steps, as shown below, lay the foundation for finding one solution. However, absolute value equations typically have two solutions, and our task is to find the one Savanah missed. We will meticulously dissect each step of the solution process, highlighting the key concepts and potential pitfalls. The importance of considering both positive and negative cases will be emphasized throughout this analysis. By the end of this article, you will have a firm grasp of how to solve absolute value equations and avoid common errors. We will also explore the underlying principles that govern these equations, providing a deeper understanding of their behavior. Absolute value, by definition, represents the distance of a number from zero, making it inherently positive. This characteristic is central to the dual-solution nature of absolute value equations. We will examine how this property influences the solution process and why it is essential to consider both positive and negative scenarios. Moreover, we will discuss various techniques for simplifying absolute value equations, including isolating the absolute value expression and applying appropriate algebraic manipulations. Mastering these techniques will empower you to tackle a wide range of absolute value problems with confidence. So, let's embark on this mathematical journey and uncover the hidden solution to Savanah's equation.

Savanah's Initial Steps

Savanah's work provides a valuable starting point for our analysis. Let's examine the steps she took to solve the equation, paying close attention to the algebraic manipulations involved. By understanding her approach, we can identify any potential omissions or errors that might have led her to find only one solution. Her work is presented as follows:

  1. 44{\frac{x}{2}+3}=8=8

  2. {\frac{x}{2}+3}$=2$

The first step involves isolating the absolute value term. Savanah correctly subtracted 3 from both sides of the original equation, resulting in 44{\frac{x}{2}+3}=8=8. This step is a fundamental algebraic manipulation, ensuring that the absolute value expression is the sole focus of the equation. Isolating the absolute value is a crucial first step in solving these types of equations. It simplifies the subsequent steps and allows us to apply the definition of absolute value more effectively. The second step involves dividing both sides of the equation by 4. This simplifies the equation further, resulting in ${\frac{x}{2}+3}=2=2. This step is also a straightforward algebraic manipulation, reducing the coefficient of the absolute value expression to 1. Simplifying the equation in this way makes it easier to analyze the two possible cases arising from the absolute value. Savanah's initial steps are accurate and demonstrate a solid understanding of basic algebraic principles. However, the key to solving absolute value equations lies in recognizing that the expression inside the absolute value bars can be either positive or negative. Savanah's work so far only represents one possibility. We need to explore the alternative scenario to find the missing solution. The absolute value of a number is its distance from zero, meaning that both a positive and a negative number can have the same absolute value. This fundamental concept is the cornerstone of solving absolute value equations. In the next section, we will delve into the two cases that arise from the absolute value expression and uncover the missing solution to Savanah's equation. By carefully considering both possibilities, we can ensure a complete and accurate solution.

The Two Cases of Absolute Value

To fully grasp the solution to this problem, it's vital to understand the core concept of absolute value. The absolute value of a number is its distance from zero, which means it's always non-negative. For example, both 3 and -3 have an absolute value of 3. This principle leads to two distinct cases when solving absolute value equations: the expression inside the absolute value bars can be equal to the positive or the negative of the value on the other side of the equation. In Savanah's case, after the initial simplification, we have ${\frac{x}{2}+3}=2=2. This equation tells us that the expression x2+3\frac{x}{2}+3 is 2 units away from zero. This can happen in two ways:

  • Case 1: The expression x2+3\frac{x}{2}+3 is equal to 2.
  • Case 2: The expression x2+3\frac{x}{2}+3 is equal to -2.

Savanah likely solved for Case 1, which is why she found one solution. To find the other solution, we need to consider Case 2. Understanding these two cases is paramount to solving absolute value equations correctly. Failing to consider both possibilities will inevitably lead to an incomplete solution. Each case represents a separate linear equation that needs to be solved independently. The solutions obtained from each case are then combined to provide the complete solution set for the absolute value equation. The absolute value function essentially splits the equation into two separate paths, each leading to a potential solution. By meticulously following each path, we can ensure that we capture all possible solutions. In the following sections, we will solve each case individually, demonstrating the algebraic manipulations involved and highlighting the differences in the solution process. Solving each case requires applying standard algebraic techniques, such as isolating the variable and performing inverse operations. However, it is crucial to remember that each case must be treated as a distinct equation, and the solutions obtained should be verified against the original absolute value equation. By carefully analyzing both cases, we can gain a comprehensive understanding of the solution set and avoid overlooking any potential answers. Let's now proceed to solve each case and unveil the missing solution to Savanah's equation.

Solving Case 1: The Positive Scenario

Let's delve into the first case, where the expression inside the absolute value bars is equal to the positive value on the other side of the equation. This scenario represents one possibility for the solution, and it's crucial to solve it accurately. In this case, we have the equation:

x2+3=2\frac{x}{2}+3=2

To solve for x, we need to isolate the variable. The first step is to subtract 3 from both sides of the equation:

x2+3−3=2−3\frac{x}{2}+3-3=2-3

This simplifies to:

x2=−1\frac{x}{2}=-1

Now, to get x by itself, we multiply both sides of the equation by 2:

2⋅x2=−1⋅22 \cdot \frac{x}{2}=-1 \cdot 2

This gives us:

x=−2x=-2

This is one solution to the absolute value equation. Savanah likely found this solution. Solving this case involves applying fundamental algebraic principles, such as subtracting and multiplying both sides of the equation by the same value. Each step is designed to isolate the variable and determine its value. It is essential to perform these operations carefully and accurately to arrive at the correct solution. The solution obtained in this case represents one of the points where the expression inside the absolute value bars equals the positive value, which satisfies the original equation. However, this is only one piece of the puzzle. To obtain the complete solution, we must also consider the second case, where the expression inside the absolute value bars equals the negative value. The importance of verifying the solution cannot be overstated. Once we obtain a potential solution, we should substitute it back into the original equation to ensure that it satisfies the equation. This step helps to identify any extraneous solutions that may have arisen during the solution process. In the next section, we will explore the second case and uncover the other solution to Savanah's equation. By comparing the solutions from both cases, we can gain a deeper understanding of the complete solution set and the nuances of solving absolute value equations.

Solving Case 2: The Negative Scenario

Now, let's tackle the second case, which often gets overlooked but is crucial for finding all solutions to an absolute value equation. In this scenario, we consider the possibility that the expression inside the absolute value bars is equal to the negative of the value on the other side of the equation. This means we have the equation:

x2+3=−2\frac{x}{2}+3=-2

Similar to Case 1, our goal is to isolate x. First, we subtract 3 from both sides:

x2+3−3=−2−3\frac{x}{2}+3-3=-2-3

This simplifies to:

x2=−5\frac{x}{2}=-5

Next, we multiply both sides by 2 to solve for x:

2⋅x2=−5⋅22 \cdot \frac{x}{2}=-5 \cdot 2

This gives us:

x=−10x=-10

This is the second solution to the absolute value equation, the one Savanah missed. This case highlights the importance of considering both positive and negative possibilities when dealing with absolute values. The negative scenario often leads to a distinct solution that is essential for completing the solution set. Solving this case involves the same algebraic principles as Case 1, but with a crucial difference: the negative sign. This difference significantly impacts the final solution and underscores the need for careful attention to detail. The solution obtained in this case represents the point where the expression inside the absolute value bars equals the negative value, which also satisfies the original equation due to the properties of absolute value. By finding both solutions, we have a complete picture of the values that make the equation true. Verifying this solution by substituting it back into the original equation is a crucial step to ensure its validity. This process confirms that the solution is not extraneous and that it accurately satisfies the equation. In the next section, we will summarize our findings and provide a comprehensive answer to the problem, emphasizing the importance of considering both cases when solving absolute value equations. By understanding the nuances of these equations, we can confidently tackle a wide range of mathematical challenges.

The Complete Solution and Key Takeaways

We've diligently explored both cases of Savanah's equation, and now we have a complete solution. We found that the equation 3+43+4{\frac{x}{2}+3}=11=11 has two solutions:

  • x=−2x = -2 (from Case 1)
  • x=−10x = -10 (from Case 2)

The other solution to the given absolute value equation that Savanah missed is x = -10. This exercise demonstrates a critical concept in solving absolute value equations: the necessity of considering both positive and negative scenarios. The absolute value function, by its very nature, introduces two possibilities, and neglecting either one can lead to an incomplete solution. Savanah's work, while accurate in its initial steps, only addressed one of these possibilities. By carefully analyzing the second case, we uncovered the missing solution and completed the puzzle. The key takeaway from this problem is the dual nature of absolute value. It represents the distance from zero, which can be achieved by both positive and negative values. When solving absolute value equations, we must always account for this duality by creating two separate cases: one where the expression inside the absolute value bars equals the positive value and another where it equals the negative value. Mastering this concept is fundamental for success in algebra and beyond. Absolute value equations appear in various mathematical contexts, and a thorough understanding of their solution strategies is essential for problem-solving. By consistently applying the two-case approach and carefully verifying the solutions, we can confidently tackle any absolute value equation that comes our way. In conclusion, Savanah's equation served as a valuable learning opportunity. It highlighted the importance of thoroughness and attention to detail when solving mathematical problems. By understanding the underlying principles and applying the appropriate techniques, we can overcome challenges and arrive at complete and accurate solutions.