Given The Function $f(x)=0.5|x-4|-3$, For What Values Of $x$ Is $f(x)=7$?A. $x=-24, X=16$ B. $x=-16, X=24$ C. $x=-1, X=9$ D. $x=1, X=-9$

Introduction

In this article, we will delve into the world of absolute value equations and explore how to solve them. Specifically, we will focus on the function $f(x)=0.5|x-4|-3$ and determine the values of $x$ for which $f(x)=7$. This problem requires a deep understanding of absolute value equations and their properties.

Understanding Absolute Value Equations

Absolute value equations are a type of equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, the absolute value of a number is always non-negative.

The general form of an absolute value equation is $|x-a|=b$, where $a$ and $b$ are constants. To solve an absolute value equation, we need to consider two cases:

• Case 1: $x-a=b$
• Case 2: $x-a=-b$

Solving the Given Function

The given function is $f(x)=0.5|x-4|-3$. We want to find the values of $x$ for which $f(x)=7$. To do this, we set the function equal to 7 and solve for $x$.

$$0.5|x-4|-3=7$$

First, we add 3 to both sides of the equation to isolate the absolute value term.

$$0.5|x-4|=10$$

Next, we multiply both sides of the equation by 2 to eliminate the fraction.

$$|x-4|=20$$

Now, we consider the two cases:

Case 1: $x-4=20$

$$x-4=20$$

$$x=24$$

Case 2: $x-4=-20$

$$x-4=-20$$

$$x=-16$$

Conclusion

In conclusion, the values of $x$ for which $f(x)=7$ are $x=-16$ and $x=24$. These values satisfy the given function and meet the condition of the problem.

Answer

The correct answer is B. $x=-16, x=24$.

Discussion

This problem requires a deep understanding of absolute value equations and their properties. The key to solving this problem is to consider the two cases and to isolate the absolute value term. By following these steps, we can determine the values of $x$ for which $f(x)=7$.

Additional Tips and Tricks

• When solving absolute value equations, always consider the two cases.
• Isolate the absolute value term by adding or subtracting the same value from both sides of the equation.
• Use the properties of absolute value to simplify the equation and solve for the variable.

Introduction

In our previous article, we explored the world of absolute value equations and solved the function $f(x)=0.5|x-4|-3$ for which $f(x)=7$. In this article, we will continue to delve into the world of absolute value equations and answer some frequently asked questions.

Q&A

Q: What is an absolute value equation?

A: An absolute value equation is a type of equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases:

• Case 1: $x-a=b$
• Case 2: $x-a=-b$

Q: What is the difference between the two cases?

A: In Case 1, the expression inside the absolute value is equal to the value outside the absolute value. In Case 2, the expression inside the absolute value is equal to the negative of the value outside the absolute value.

Q: How do I isolate the absolute value term?

A: To isolate the absolute value term, you need to add or subtract the same value from both sides of the equation. This will help you to simplify the equation and solve for the variable.

Q: What are some common mistakes to avoid when solving absolute value equations?

A: Some common mistakes to avoid when solving absolute value equations include:

• Not considering both cases
• Not isolating the absolute value term
• Not checking for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original equation and verify that it is true.

Q: What are some real-world applications of absolute value equations?

A: Absolute value equations have many real-world applications, including:

• Physics: to model the motion of objects
• Engineering: to design and optimize systems
• Economics: to model economic systems and make predictions

Conclusion

In conclusion, absolute value equations are a powerful tool for solving problems in mathematics and other fields. By understanding the properties of absolute value and how to solve absolute value equations, you can tackle a wide range of problems and make predictions with confidence.

Additional Tips and Tricks

• Practice, practice, practice: the more you practice solving absolute value equations, the more comfortable you will become with the process.
• Use visual aids: graphs and charts can help you to visualize the problem and understand the solution.
• Check your work: always check your work to make sure that the solution is correct.

By following these tips and tricks, you can become proficient in solving absolute value equations and tackle more complex problems with confidence.