Savanah's Solution To Absolute Value Equation 3+4|x/2+3|=11 A Step By Step Guide

In this article, we delve into Savanah's method for solving the absolute value equation $3+4 mfrac{x}{2}+3 =11$. We'll meticulously examine each step, providing a comprehensive explanation to ensure a clear understanding of the process. Our focus will be on validating Savanah's approach and highlighting key concepts in solving absolute value equations. We will explore potential alternative methods and discuss the importance of checking solutions in the context of absolute value problems. This analysis aims to serve as a valuable resource for students and educators alike, offering insights into the intricacies of algebraic problem-solving.

Understanding the Initial Equation

Absolute value equations can sometimes seem tricky, but they become much more manageable when broken down into steps. Savanah starts with the equation $3+4 mfrac{x}{2}+3 =11$. The key to solving such equations lies in isolating the absolute value term first. This involves performing algebraic manipulations to get the absolute value expression by itself on one side of the equation. Think of it as peeling away the layers to get to the core of the problem. In this specific case, the absolute value term is $ mfrac{x}{2}+3 $, and our goal is to isolate this term before addressing the absolute value itself. This initial setup is crucial because it sets the stage for the subsequent steps, where we'll deal with the two possible scenarios arising from the absolute value.

Step 1 Isolating the Absolute Value

In the initial step, Savanah correctly isolates the absolute value term. Savanah's first step is to subtract 3 from both sides of the equation. This is a fundamental algebraic operation, ensuring that the equation remains balanced. By subtracting 3 from both sides, we eliminate the constant term on the left side, bringing us closer to isolating the absolute value expression. The equation now becomes $4 mfrac{x}{2}+3 =8$. This step is vital as it simplifies the equation and allows us to focus solely on the absolute value expression. Think of it as clearing away the clutter to see the main subject more clearly. By isolating the absolute value, we pave the way for the next step, where we will divide to further simplify the equation.

Step 2 Simplifying the Absolute Value Expression

Savanah's next move is to divide both sides of the equation by 4. This step further isolates the absolute value expression. Dividing both sides by 4 maintains the equation's balance while simplifying the coefficient of the absolute value term. This results in the equation $ mfrac{x}{2}+3 =2$. This is a crucial simplification because it prepares us to address the two possible cases that arise from the absolute value. Recall that the absolute value of a number is its distance from zero, meaning it can be either the number itself or its negative. This step sets the stage for considering both possibilities, ensuring we find all potential solutions to the equation.

Step 3 Addressing the Absolute Value Two Cases

Here's where the heart of the problem lies dealing with the absolute value. The absolute value of an expression can be equal to a certain value if the expression itself is equal to that value or if the expression is equal to the negative of that value. This is a fundamental property of absolute values. Therefore, the equation $ mfrac{x}{2}+3 =2$ gives rise to two separate cases. Case 1 is when $ mfrac{x}{2}+3=2$ and Case 2 is when $ mfrac{x}{2}+3=-2$. Savanah correctly identifies and sets up the first case. By recognizing these two possibilities, we ensure that we consider all potential solutions to the original equation. Failing to address both cases would lead to an incomplete solution.

Solving Case 1 and Finding One Solution

Now, let's solve the first case presented by Savanah: $ mfrac{x}{2}+3=2$. To solve for x, we first subtract 3 from both sides of the equation. This isolates the term with x on one side. This yields $ mfrac{x}{2}=-1$. Then, we multiply both sides of the equation by 2 to solve for x. This gives us $x=-2$. Savanah has found one solution to the original equation. It's important to note that this is only one possible solution because we have yet to consider the second case arising from the absolute value. Finding this solution is a significant step, but it's not the end of the journey.

Addressing the Second Case A Complete Solution

To obtain the complete solution set, we must now consider the second case: $ mfrac{x}{2}+3=-2$. This case arises from the definition of absolute value, where the expression inside the absolute value can be either positive or negative. To solve this case, we follow a similar procedure as before. First, we subtract 3 from both sides of the equation, which gives us $ mfrac{x}{2}=-5$. Then, we multiply both sides by 2 to isolate x, resulting in $x=-10$. This is the second solution to the original equation. By considering both cases, we have now found the complete set of solutions.

Verifying the Solutions Accuracy is Key

It is always crucial to verify the solutions obtained when dealing with absolute value equations. This is because extraneous solutions can sometimes arise due to the nature of absolute values. To verify, we substitute each solution back into the original equation and check if the equation holds true. Let's start with the first solution, $x=-2$. Substituting this into the original equation, $3+4 mfrac{x}{2}+3 =11$, we get $3+4 mfrac{-2}{2}+3 =3+4 -1+3 =3+4 2 =3+8=11$, which is true. Now let's check the second solution, $x=-10$. Substituting this into the original equation, we get $3+4 mfrac{-10}{2}+3 =3+4 -5+3 =3+4 -2 =3+4(2)=3+8=11$, which is also true. Therefore, both $x=-2$ and $x=-10$ are valid solutions to the equation.

Alternative Methods for Solving Absolute Value Equations Different Approaches

While Savanah's method is a standard and effective way to solve absolute value equations, there might be alternative approaches depending on the complexity of the equation. One approach could involve graphing the equation and finding the points of intersection. Another approach might involve using properties of absolute values to simplify the equation before solving. Exploring these alternative methods can provide a deeper understanding of absolute value equations and enhance problem-solving skills. Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation and the solver's preference. For instance, graphing can be particularly useful for visualizing the solutions, while algebraic methods are generally more efficient for solving equations.

Common Mistakes to Avoid Absolute Value Pitfalls

When solving absolute value equations, there are several common mistakes that students often make. One common mistake is forgetting to consider both cases arising from the absolute value. Another mistake is incorrectly applying algebraic operations, such as adding or subtracting terms from only one side of the equation. It's also important to correctly simplify the equation and avoid errors in arithmetic. Being mindful of these potential pitfalls can help students avoid mistakes and solve absolute value equations accurately. Regularly reviewing the steps and concepts involved in solving these equations can also help reinforce understanding and prevent errors.

Conclusion Mastering Absolute Value Equations

In summary, Savanah's approach to solving the absolute value equation is correct and follows the standard procedure for such problems. By isolating the absolute value term, considering both positive and negative cases, and verifying the solutions, Savanah successfully found one solution. However, to fully solve the equation, the second case must also be considered. This detailed analysis provides a thorough understanding of the steps involved in solving absolute value equations and highlights the importance of careful and methodical problem-solving. Mastering these concepts is crucial for success in algebra and beyond. This comprehensive guide should serve as a valuable resource for students and educators alike, promoting a deeper understanding of absolute value equations.