Solving Absolute Value Equations A Step-by-Step Guide

Hey guys! Let's break down how to solve the equation $3 \left|\frac{x}{3}-4\right|=12$. This problem involves absolute values, which might seem tricky, but don't worry, we'll tackle it step by step. Absolute value equations basically mean we have two possibilities to consider, so let’s dive in!

Understanding Absolute Value

Before we jump into solving, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero. So, $|5|$ is 5, and $|-5|$ is also 5. Absolute value always gives us a non-negative result. When we have an equation with absolute values, it means the expression inside the absolute value bars could be either positive or negative, but the absolute value of that expression must equal the given value.

In our equation, $3\left|\frac{x}{3}-4\right|=12$, the absolute value part is $\left|\frac{x}{3}-4\right|$. This means the expression $\frac{x}{3}-4$ could be either a positive number or a negative number, but its absolute value is what we're really concerned with. This is crucial because it leads us to consider two separate cases, ensuring we capture all possible solutions for $x$. Think of it like this: if $|y| = 5$, then $y$ could be either 5 or -5. We apply the same logic to the expression inside our absolute value.

When faced with absolute value equations, it's essential to isolate the absolute value term first. This simplifies the problem and makes it easier to handle the two possible cases. In our equation, the absolute value term is already somewhat isolated, but we need to get rid of that 3 that's multiplying the absolute value. This involves a simple division, which is the first step in our solving process. Once the absolute value is isolated, we can then split the equation into two separate equations, one where the expression inside the absolute value is positive and one where it's negative. This approach is a systematic way to handle absolute value equations and ensures we don't miss any potential solutions. Remember, absolute value problems are just about considering both positive and negative scenarios, so breaking it down into two cases makes it much more manageable.

Step-by-Step Solution

1. Isolate the Absolute Value

Our equation is $3\left|\frac{x}{3}-4\right|=12$. To isolate the absolute value, we need to get rid of the 3 that's multiplying it. We can do this by dividing both sides of the equation by 3:

$\left|\frac{x}{3}-4\right| = \frac{12}{3}$

$\left|\frac{x}{3}-4\right| = 4$

Now we have the absolute value isolated, which is exactly what we wanted! This simplifies the problem significantly and sets us up perfectly for the next step. Isolating the absolute value is a critical first move because it allows us to clearly see what expression's distance from zero we're dealing with. Without this step, it would be much harder to proceed and correctly split the problem into two cases. It’s like setting the stage for the main act – you need a clear stage to perform well. So, always remember, when you see an absolute value, your first goal is to get it alone on one side of the equation. This one step makes the rest of the solution much smoother and more intuitive, allowing us to focus on the core concept of absolute value: considering both positive and negative possibilities.

2. Split into Two Cases

Since the absolute value of $\frac{x}{3}-4$ is 4, this means that $\frac{x}{3}-4$ could be either 4 or -4. This is the fundamental concept of absolute values in action. The expression inside the absolute value bars has two potential identities that satisfy the equation, making it a bit more complex than a regular algebraic equation. To tackle this, we create two separate equations, each representing one of these possibilities:

Case 1: $\frac{x}{3}-4 = 4$

Case 2: $\frac{x}{3}-4 = -4$

By splitting the problem into these two cases, we transform one absolute value equation into two linear equations, which are much easier to solve. Each case represents a potential reality for the value of $x$, and we need to solve both to find all possible solutions. This approach is a cornerstone of solving absolute value equations, allowing us to systematically address the inherent dual nature of the absolute value. It's like having a fork in the road – we need to explore both paths to find all the destinations. By addressing each case separately, we ensure that we don't miss any solutions and that we fully account for the properties of absolute value.

3. Solve Case 1: $\frac{x}{3}-4 = 4$

To solve this equation, we first want to get rid of the -4 on the left side. We can do this by adding 4 to both sides:

$\frac{x}{3} = 4 + 4$

$\frac{x}{3} = 8$

Now, to isolate $x$, we need to get rid of the division by 3. We do this by multiplying both sides by 3:

$x = 8 imes 3$

$x = 24$

So, one possible solution is $x = 24$. This case demonstrates the straightforward application of algebraic principles to solve for $x$. We systematically reverse the operations affecting $x$, first dealing with the subtraction and then the division. Each step is aimed at isolating $x$, and the logic is consistent with solving any linear equation. This process highlights the importance of maintaining balance in an equation – whatever operation we perform on one side, we must perform on the other. This ensures that the equality remains valid and that we arrive at the correct solution. The solution $x = 24$ represents one of the values that, when substituted back into the original absolute value equation, will satisfy the condition. It's a crucial piece of the puzzle, but we're not done yet – we still have Case 2 to solve.

4. Solve Case 2: $\frac{x}{3}-4 = -4$

Again, we start by adding 4 to both sides to get rid of the -4 on the left:

$\frac{x}{3} = -4 + 4$

$\frac{x}{3} = 0$

Next, we multiply both sides by 3 to isolate $x$:

$x = 0 imes 3$

$x = 0$

So, our second possible solution is $x = 0$. This case further illustrates the methodical approach to solving for $x$. As in Case 1, we systematically undo the operations affecting $x$, ensuring that we maintain the equality of the equation. The arithmetic in this case is particularly straightforward, but the principle remains the same. Solving Case 2 is just as important as solving Case 1 because it gives us the other value of $x$ that satisfies the absolute value equation. Without considering both cases, we would only have a partial solution, and in the context of absolute value equations, it's crucial to capture all possible solutions. The result $x = 0$ is another critical piece of the puzzle, and together with $x = 24$, it gives us a complete solution set.

5. Check Your Solutions

It's always a good idea to check our solutions to make sure they work in the original equation. This is especially important with absolute value equations because sometimes we might introduce extraneous solutions (solutions that don't actually work). Let's check our solutions:

For $x = 24$:

$3\left|\frac{24}{3}-4\right| = 3|8-4| = 3|4| = 3 imes 4 = 12$ (This works!)

For $x = 0$:

$3\left|\frac{0}{3}-4\right| = 3|0-4| = 3|-4| = 3 imes 4 = 12$ (This also works!)

Both solutions check out, which is awesome! Verifying solutions is a fundamental step in the problem-solving process, particularly when dealing with absolute value equations. It’s like the final seal of approval, ensuring that our hard work has paid off and that we haven’t made any algebraic missteps along the way. The process of checking involves substituting each potential solution back into the original equation and seeing if it holds true. This step not only confirms the correctness of our solutions but also enhances our understanding of the equation itself. In the context of absolute value equations, checking is even more critical because the nature of absolute values can sometimes lead to extraneous solutions – values that appear to be solutions but don't actually satisfy the original equation. By taking the time to check, we can confidently present our solutions, knowing that they are accurate and valid.

Final Answer

The solutions to the equation $3\left|\frac{x}{3}-4\right|=12$ are $x = 24$ and $x = 0$.

Conclusion

So there you have it! Solving absolute value equations involves isolating the absolute value, splitting the problem into two cases, solving each case separately, and then checking your solutions. It might seem like a lot of steps, but once you get the hang of it, it becomes much easier. Remember, the key is to break the problem down and tackle it step by step. Keep practicing, and you'll become a pro at solving these types of equations!