Solve 2|x+6|-4 = 6

$$

Hey guys! Today, we're diving deep into a cool problem involving absolute value functions. We've got this function, $f(x) = 2|x+6| - 4$, and the big question is: for what values of $x$ is $f(x) = 6$? This is a classic algebra challenge, and by the end of this, you'll be a pro at tackling these kinds of equations. We're going to break it down step-by-step, making sure you understand every bit of the process. So, grab your notebooks, maybe a snack, and let's get this math party started! We'll be looking at the properties of absolute value, how it affects our equation, and how to isolate our variable, $x$, to find all possible solutions. Remember, absolute value means the distance from zero, so it can be a bit tricky sometimes, but we've got this.

Understanding the Absolute Value Function

Before we jump into solving $f(x)=6$, let's quickly refresh our memory on what absolute value actually means. The absolute value of a number, denoted by vertical bars like $|a|$, is its distance from zero on the number line. This means that the absolute value of any number is always non-negative. For instance, $|5| = 5$ and $|-5| = 5$. When we see $|x+6|$ in our function $f(x) = 2|x+6| - 4$, it represents the distance between $x$ and $-6$ on the number line. This non-negative property is super crucial because it means the expression inside the absolute value, $(x+6)$, can be either positive or negative, and we'll get the same positive result when we take its absolute value. This is exactly why absolute value equations often have two possible solutions. For our problem, we're setting $f(x)$ equal to 6, which is a positive number. This is a good sign, as it suggests we're likely to find real solutions. If we were asked to solve for $f(x)$ equal to a negative number, we'd immediately know there are no real solutions because the absolute value part, $2|x+6|$, will always be greater than or equal to zero, making $f(x)$ always greater than or equal to $-4$. But since we're dealing with $f(x)=6$, we're in business!

Setting Up the Equation

Alright, the first step in solving for $x$ when $f(x)=6$ is to substitute the given function into the equation. So, we have:

$2|x+6| - 4 = 6$

Our main goal here is to isolate the absolute value term, $|x+6|$. To do this, we'll perform operations on both sides of the equation, just like we would with any regular algebraic equation. Think of it as peeling back the layers of the function to get to the core variable, $x$. First, we need to get rid of that '-4'. We can do this by adding 4 to both sides of the equation:

$2|x+6| - 4 + 4 = 6 + 4$

This simplifies to:

$2|x+6| = 10$

Now, the absolute value term is being multiplied by 2. To isolate $|x+6|$, we'll divide both sides of the equation by 2:

rac{2|x+6|}{2} = rac{10}{2}

Which gives us:

$|x+6| = 5$

This is the key simplified equation we need to solve. It tells us that the expression $(x+6)$ must have a distance of 5 from zero. Remember, distance is always positive. This is where the two possible scenarios for absolute value come into play.

Solving for x: The Two Cases

Now that we have $|x+6| = 5$, we need to consider the two possibilities that arise from the definition of absolute value. Essentially, the expression inside the absolute value, $(x+6)$, can either be equal to 5 or equal to -5, because both 5 and -5 have a distance of 5 from zero.

Case 1: $x+6$ is positive (or zero)

In this case, the expression inside the absolute value is non-negative, so we can simply remove the absolute value bars:

$x+6 = 5$

To solve for $x$, we subtract 6 from both sides:

$x = 5 - 6$

$x = -1$

So, one potential solution is $x = -1$. Let's keep this in our back pocket!

Case 2: $x+6$ is negative

In this case, the expression inside the absolute value is negative. To make it positive (which is what the absolute value does), we can think of it as $-(x+6)$ equaling 5, or simply set $(x+6)$ equal to the negative counterpart of 5:

$x+6 = -5$

To solve for $x$ here, we again subtract 6 from both sides:

$x = -5 - 6$

$x = -11$

And there we have our second potential solution: $x = -11$. So, it looks like we have two values for $x$ that satisfy the original equation.

Verifying the Solutions

It's always a good practice, especially with absolute value equations, to check our answers by plugging them back into the original function $f(x) = 2|x+6| - 4$. This helps us confirm that we haven't made any mistakes along the way.

Checking $x = -1$:

Let's plug $x = -1$ into the function:

$f(-1) = 2|-1+6| - 4$

$f(-1) = 2|5| - 4$

Since $|5| = 5$, we have:

$f(-1) = 2(5) - 4$

$f(-1) = 10 - 4$

$f(-1) = 6$

Perfect! $x = -1$ is indeed a correct solution because it makes $f(x)$ equal to 6.

Checking $x = -11$:

Now let's plug $x = -11$ into the function:

$f(-11) = 2|-11+6| - 4$

$f(-11) = 2|-5| - 4$

Since $|-5| = 5$, we have:

$f(-11) = 2(5) - 4$

$f(-11) = 10 - 4$

$f(-11) = 6$

Awesome! $x = -11$ also makes $f(x)$ equal to 6. So, this solution is also correct.

Conclusion: The Final Answer

We've successfully navigated the world of absolute value equations! By setting up the equation $2|x+6| - 4 = 6$, isolating the absolute value term to get $|x+6| = 5$, and then considering both the positive and negative cases for the expression inside the absolute value, we found two distinct values for $x$. Both $x = -1$ and $x = -11$ satisfy the condition $f(x) = 6$. Therefore, the values of $x$ for which $f(x)=6$ are $x=-1$ and $x=-11$. This matches option B from the given choices. Keep practicing these types of problems, guys, and you'll master them in no time! Remember, the key is to break down the problem, understand the properties of absolute value, and always check your work. Happy problem-solving!