Given The Function $f(x)=2|x+6|-4$, For What Values Of $x$ Is $f(x)=6$?A. $x=-1, X=11$ B. $x=-1, X=-11$ C. $x=14, X=-26$ D. $x=26, X=-14$

Introduction

In mathematics, absolute value functions are a crucial concept in algebra and calculus. They are used to model real-world problems, such as distance, temperature, and financial applications. In this article, we will focus on solving the absolute value function $f(x)=2|x+6|-4$ to find the values of $x$ for which $f(x)=6$. We will break down the solution step by step, using algebraic techniques and absolute value properties.

Understanding the Absolute Value Function

The absolute value function $f(x)=2|x+6|-4$ is a piecewise function, meaning it has different expressions for different intervals of $x$. The absolute value expression $|x+6|$ is the key to understanding this function. When $x+6$ is non-negative, $|x+6|=x+6$, and when $x+6$ is negative, $|x+6|=-(x+6)$.

Case 1: When x+6 ≥ 0

When $x+6 \geq 0$, we can rewrite the absolute value function as $f(x)=2(x+6)-4$. Simplifying this expression, we get $f(x)=2x+12-4$, which further simplifies to $f(x)=2x+8$.

Case 2: When x+6 < 0

When $x+6 < 0$, we can rewrite the absolute value function as $f(x)=2(-(x+6))-4$. Simplifying this expression, we get $f(x)=-2(x+6)-4$, which further simplifies to $f(x)=-2x-12-4$, and finally to $f(x)=-2x-16$.

Setting Up the Equation

We want to find the values of $x$ for which $f(x)=6$. Using the expressions we derived earlier, we can set up two equations:

1. For $x+6 \geq 0$: $2x+8=6$
2. For $x+6 < 0$: $-2x-16=6$

Solving the Equations

Solving Equation 1: 2x + 8 = 6

Subtracting 8 from both sides, we get $2x=-2$. Dividing both sides by 2, we get $x=-1$.

Solving Equation 2: -2x - 16 = 6

Adding 16 to both sides, we get $-2x=22$. Dividing both sides by -2, we get $x=-11$.

Conclusion

In conclusion, we have found the values of $x$ for which $f(x)=6$. The solutions are $x=-1$ and $x=-11$. These values satisfy the conditions for both cases, and we have verified them using algebraic techniques.

Final Answer

The final answer is: $\boxed{B}$

Introduction

In our previous article, we solved the absolute value function $f(x)=2|x+6|-4$ to find the values of $x$ for which $f(x)=6$. We broke down the solution step by step, using algebraic techniques and absolute value properties. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into solving absolute value functions.

Q: What is the absolute value function, and why is it important?

A: The absolute value function is a mathematical function that returns the distance of a number from zero on the number line. It is denoted by the symbol $|x|$, and it is used to model real-world problems, such as distance, temperature, and financial applications. The absolute value function is important because it helps us to solve equations and inequalities that involve absolute values.

Q: What are the two cases for the absolute value function?

A: The two cases for the absolute value function are:

1. When $x+6 \geq 0$, we can rewrite the absolute value function as $f(x)=2(x+6)-4$.
2. When $x+6 < 0$, we can rewrite the absolute value function as $f(x)=2(-(x+6))-4$.

Q: How do we solve the absolute value function?

A: To solve the absolute value function, we need to consider both cases and set up two equations:

1. For $x+6 \geq 0$: $2x+8=6$
2. For $x+6 < 0$: $-2x-16=6$

We then solve each equation separately to find the values of $x$ that satisfy the equation.

Q: What are the values of $x$ for which $f(x)=6$?

A: The values of $x$ for which $f(x)=6$ are $x=-1$ and $x=-11$. These values satisfy the conditions for both cases, and we have verified them using algebraic techniques.

Q: How do we know which case to use?

A: To determine which case to use, we need to check the sign of $x+6$. If $x+6 \geq 0$, we use Case 1. If $x+6 < 0$, we use Case 2.

Q: What if the equation has multiple solutions?

A: If the equation has multiple solutions, we need to check each solution to see if it satisfies the conditions for both cases. If a solution satisfies both cases, it is a valid solution.

Q: Can we use other methods to solve the absolute value function?

A: Yes, we can use other methods to solve the absolute value function, such as graphing or using a calculator. However, algebraic techniques are often the most efficient and effective way to solve absolute value functions.

Conclusion

In conclusion, solving absolute value functions requires careful consideration of both cases and the use of algebraic techniques. By following the steps outlined in this article, you can solve absolute value functions and find the values of $x$ that satisfy the equation.

Final Answer

The final answer is: $\boxed{B}$