Given The Function F ( X ) = 4 ∣ X − 5 ∣ + 3 F(x) = 4|x - 5| + 3 F ( X ) = 4∣ X − 5∣ + 3 , For What Values Of X X X Is F ( X ) = 15 F(x) = 15 F ( X ) = 15 ?A. X = 2 , X = 8 X = 2, X = 8 X = 2 , X = 8 B. X = 1.5 , X = 8 X = 1.5, X = 8 X = 1.5 , X = 8 C. X = 2 , X = 7.5 X = 2, X = 7.5 X = 2 , X = 7.5 D. X = 0.5 , X = 7.5 X = 0.5, X = 7.5 X = 0.5 , X = 7.5

Introduction

In mathematics, absolute value equations are a type of equation that involves the absolute value of a variable or expression. These equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving absolute value equations, specifically the function $f(x) = 4|x - 5| + 3$, and find the values of $x$ for which $f(x) = 15$.

Understanding Absolute Value Equations

Absolute value equations involve the absolute value of a variable or expression, which is denoted by the symbol $|x|$. The absolute value of a number is its distance from zero on the number line, without considering its direction. For example, the absolute value of $-3$ is $3$, and the absolute value of $3$ is also $3$.

The Function $f(x) = 4|x - 5| + 3$

The given function is $f(x) = 4|x - 5| + 3$. This function involves the absolute value of the expression $x - 5$. To solve for $x$, we need to isolate the absolute value expression and then solve for the variable.

Step 1: Isolate the Absolute Value Expression

The first step is to isolate the absolute value expression by subtracting $3$ from both sides of the equation:

$f(x) = 15$

Subtracting $3$ from both sides gives us:

$4|x - 5| = 12$

Step 2: Divide Both Sides by 4

Next, we divide both sides of the equation by $4$ to isolate the absolute value expression:

$|x - 5| = 3$

Step 3: Solve for $x$

Now that we have isolated the absolute value expression, we can solve for $x$. To do this, we need to consider two cases: when $x - 5$ is positive, and when $x - 5$ is negative.

Case 1: $x - 5 \geq 0$

When $x - 5 \geq 0$, we can remove the absolute value sign and solve for $x$:

$x - 5 = 3$

Adding $5$ to both sides gives us:

$x = 8$

Case 2: $x - 5 < 0$

When $x - 5 < 0$, we can remove the absolute value sign and solve for $x$:

$-(x - 5) = 3$

Simplifying the equation gives us:

$-x + 5 = 3$

Subtracting $5$ from both sides gives us:

$-x = -2$

Multiplying both sides by $-1$ gives us:

$x = 2$

Conclusion

In conclusion, we have solved the absolute value equation $f(x) = 4|x - 5| + 3$ and found the values of $x$ for which $f(x) = 15$. The solutions are $x = 2$ and $x = 8$. Therefore, the correct answer is:

A. $x = 2, x = 8$

Final Answer

Introduction

In our previous article, we discussed how to solve absolute value equations, specifically the function $f(x) = 4|x - 5| + 3$, and found the values of $x$ for which $f(x) = 15$. In this article, we will provide a Q&A guide to help you better understand and solve absolute value equations.

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 its direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to isolate the absolute value expression and then solve for the variable. You can do this by using the following steps:

1. Isolate the absolute value expression by subtracting or adding a constant to both sides of the equation.
2. Divide both sides of the equation by a coefficient, if necessary.
3. Consider two cases: when the expression inside the absolute value is positive, and when it is negative.
4. Solve for the variable in each case.

Q: What is the difference between $|x|$ and $x$?

A: The main difference between $|x|$ and $x$ is that $|x|$ represents the absolute value of $x$, which is its distance from zero on the number line, without considering its direction. On the other hand, $x$ represents the value of $x$ itself, without considering its sign.

Q: How do I know which case to use when solving an absolute value equation?

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, you can remove the absolute value sign and solve for the variable. If the expression is negative, you need to remove the absolute value sign and multiply the expression by $-1$ to get a positive value.

Q: What is the significance of the absolute value in an equation?

A: The absolute value in an equation represents the distance of the variable from a certain point on the number line. It can be used to model real-world problems, such as the distance between two points, the magnitude of a vector, or the absolute error in a measurement.

Q: Can I use absolute value equations to model real-world problems?

A: Yes, absolute value equations can be used to model real-world problems. For example, you can use absolute value equations to model the distance between two points, the magnitude of a vector, or the absolute error in a measurement.

Q: What are some common applications of absolute value equations?

A: Some common applications of absolute value equations include:

• Modeling the distance between two points
• Modeling the magnitude of a vector
• Modeling the absolute error in a measurement
• Modeling the cost of a product or service
• Modeling the time it takes to complete a task

Conclusion

In conclusion, absolute value equations are a powerful tool for modeling real-world problems. By understanding how to solve absolute value equations, you can apply them to a wide range of applications, from science and engineering to finance and economics. We hope this Q&A guide has helped you better understand and solve absolute value equations.

Final Answer

The final answer is that absolute value equations are a valuable tool for modeling real-world problems, and by understanding how to solve them, you can apply them to a wide range of applications.