Solving F(x) = 18 For The Function F(x) = 3|x-2|+6
In this article, we will delve into the process of finding the values of x for which the function f(x) = 3|x-2| + 6 equals 18. This involves understanding absolute value functions and applying algebraic techniques to solve the resulting equations. The key to solving this problem lies in recognizing that the absolute value function introduces two possibilities, which we must address separately to find all possible solutions for x. This exploration will not only enhance your problem-solving skills but also deepen your understanding of how absolute value functions behave within equations. Let's begin by understanding the function and its properties before diving into the solution.
The function we are working with is f(x) = 3|x-2| + 6. The absolute value portion, |x-2|, is crucial because it transforms any input into its non-negative counterpart. This means that whatever the expression inside the absolute value evaluates to, the result will always be either zero or a positive number. The presence of the absolute value creates a piecewise nature to the function, which means we need to consider two separate cases to solve the equation f(x) = 18. Understanding this fundamental property is essential for correctly solving equations involving absolute values. The function f(x) essentially takes the absolute value of (x-2), multiplies it by 3, and then adds 6. This transformation affects the graph of the function, making it V-shaped, with the vertex determined by the expression inside the absolute value. This vertex plays a critical role in identifying the solutions to the equation.
The goal is to find the values of x that make f(x) equal to 18. Essentially, we're trying to solve the equation 3|x-2| + 6 = 18. This requires us to isolate the absolute value term and then consider the two possible scenarios arising from the absolute value: the expression inside the absolute value is either positive or negative. Each of these scenarios will lead to a different linear equation that needs to be solved independently. By addressing both possibilities, we ensure that we capture all values of x that satisfy the original equation. This method is a standard approach when dealing with absolute value equations and forms the core of the solution process.
To solve for the values of x when f(x) = 18, we first need to set up the equation. Given the function f(x) = 3|x-2| + 6, we substitute 18 for f(x), resulting in the equation:
3|x-2| + 6 = 18
This is our starting point. The next step involves isolating the absolute value term to simplify the equation and make it easier to solve. This isolation is crucial because it allows us to deal with the absolute value separately, which is essential for handling the two potential scenarios that arise due to the absolute value function.
To isolate the absolute value term, we need to perform some algebraic manipulations. The first step is to subtract 6 from both sides of the equation. This operation maintains the equality and moves us closer to having the absolute value term by itself on one side of the equation. Subtracting 6 from both sides gives us:
3|x-2| = 18 - 6
Simplifying the right side of the equation, we get:
3|x-2| = 12
Now, we need to further isolate the absolute value by dividing both sides of the equation by 3. This step is essential to get the absolute value term alone, which is necessary for addressing the two possible cases. Dividing both sides by 3, we have:
|x-2| = 12 / 3
Simplifying the right side again, we arrive at the isolated absolute value equation:
|x-2| = 4
This equation is the key to unlocking the solutions for x. With the absolute value term isolated, we are now ready to consider the two possible cases that arise from the definition of the absolute value.
The first case we need to consider is when the expression inside the absolute value, (x-2), is either positive or zero. In this scenario, the absolute value does not change the expression, and we can simply remove the absolute value signs. This leads us to the equation:
x - 2 = 4
To solve for x in this case, we need to isolate x on one side of the equation. We can do this by adding 2 to both sides of the equation. This operation maintains the equality and allows us to solve for x directly. Adding 2 to both sides gives us:
x = 4 + 2
Simplifying the right side, we find the first solution for x:
x = 6
This is one of the potential values of x that makes f(x) = 18. However, we need to remember that the absolute value introduces another possibility, which we will explore in the next section. It is important to recognize that this solution is valid only under the condition that (x-2) is positive or zero, which it is in this case since 6 - 2 = 4, which is a positive number. This check is a good practice to ensure that our solutions are consistent with the initial conditions imposed by the absolute value.
Now, let's consider the second case, which occurs when the expression inside the absolute value, (x-2), is negative. In this scenario, the absolute value function will change the sign of the expression, making it positive. This means that |x-2| becomes -(x-2). So, our equation transforms to:
-(x - 2) = 4
To solve this equation, we first need to distribute the negative sign across the terms inside the parentheses. This gives us:
-x + 2 = 4
Now, we want to isolate x. The first step is to subtract 2 from both sides of the equation, which maintains the equality and moves us closer to isolating x. Subtracting 2 from both sides, we get:
-x = 4 - 2
Simplifying the right side, we have:
-x = 2
Finally, to solve for x, we multiply both sides of the equation by -1. This changes the sign of both sides and gives us the value of x:
x = -2
This is the second potential solution for x that makes f(x) = 18. As with the first case, it's a good practice to check if this solution is consistent with the initial condition that (x-2) is negative. In this case, -2 - 2 = -4, which is indeed a negative number, so this solution is valid.
By considering both cases, we have found two values of x that satisfy the equation f(x) = 18, where f(x) = 3|x-2| + 6. These values are:
- x = 6
- x = -2
Therefore, the solutions to the equation are x = 6 and x = -2. These are the only two values of x that will make the function f(x) equal to 18. It's crucial to remember that when dealing with absolute value equations, it's necessary to consider both the positive and negative cases of the expression inside the absolute value to ensure that all possible solutions are found.
To verify these solutions, we can substitute them back into the original equation f(x) = 3|x-2| + 6 and check if the result is indeed 18. Let's start with x = 6:
f(6) = 3|6-2| + 6 = 3|4| + 6 = 3(4) + 6 = 12 + 6 = 18
As we can see, when x = 6, f(x) equals 18, so this solution is correct. Now let's verify the second solution, x = -2:
f(-2) = 3|-2-2| + 6 = 3|-4| + 6 = 3(4) + 6 = 12 + 6 = 18
Again, when x = -2, f(x) equals 18, confirming that this solution is also correct. This verification step is an excellent way to ensure the accuracy of your solutions and to reinforce your understanding of how the absolute value function works.
In conclusion, we have successfully determined the values of x for which the function f(x) = 3|x-2| + 6 equals 18. By carefully considering the two cases arising from the absolute value function, we found that the solutions are x = 6 and x = -2. This process involved isolating the absolute value term, setting up two separate equations based on the sign of the expression inside the absolute value, and solving each equation independently. This method is a fundamental technique in algebra for solving equations involving absolute values.
Understanding how to solve absolute value equations is crucial in mathematics as they appear in various contexts, including calculus, linear algebra, and real analysis. The ability to break down an absolute value equation into separate cases and solve each case individually is a valuable skill. Moreover, verifying the solutions by substituting them back into the original equation is a best practice that helps ensure accuracy and deepens understanding.
This exploration not only provides the specific solutions to the given problem but also reinforces the broader concepts of absolute value functions and algebraic problem-solving techniques. By mastering these techniques, you can confidently tackle more complex problems involving absolute values and other types of functions. Remember, the key to success in mathematics lies in a solid understanding of fundamental principles and the ability to apply them methodically and accurately.
