For Which Equations Is $x = 9$ A Possible Solution? Check All That Apply.- ∣ X ∣ = 9 |x| = 9 ∣ X ∣ = 9 - − ∣ X ∣ = 9 -|x| = 9 − ∣ X ∣ = 9 - − ∣ − X ∣ = 9 -|-x| = 9 − ∣ − X ∣ = 9 - − ∣ − X ∣ = − 9 -|-x| = -9 − ∣ − X ∣ = − 9 - ∣ X ∣ = − 9 |x| = -9 ∣ X ∣ = − 9 - ∣ − X ∣ = 9 |-x| = 9 ∣ − X ∣ = 9 - ∣ − X ∣ = − 9 |-x| = -9 ∣ − X ∣ = − 9

Introduction

Absolute value equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. In this article, we will explore the possibilities of the equation $x = 9$ as a solution to various absolute value equations. We will examine each option carefully and determine whether $x = 9$ is a possible solution.

Understanding Absolute Value

Before we dive into the equations, let's take a moment to understand what absolute value means. The absolute value of a number is its distance from zero on the number line. In other words, it is the value of the number without considering its sign. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is also $4$.

Option 1: $|x| = 9$

Let's start with the first option: $|x| = 9$. To solve this equation, we need to find the values of $x$ that make the absolute value of $x$ equal to $9$. Since the absolute value of $x$ is $9$, we know that $x$ can be either $9$ or $-9$. Therefore, $x = 9$ is a possible solution to this equation.

Option 2: $-|x| = 9$

Now, let's consider the second option: $-|x| = 9$. To solve this equation, we need to find the values of $x$ that make the negative of the absolute value of $x$ equal to $9$. Since the negative of the absolute value of $x$ is $-9$, we know that $x$ can be either $-9$ or $9$. Therefore, $x = 9$ is a possible solution to this equation.

Option 3: $-|-x| = 9$

Next, let's examine the third option: $-|-x| = 9$. To solve this equation, we need to find the values of $x$ that make the negative of the absolute value of $-x$ equal to $9$. Since the negative of the absolute value of $-x$ is $-9$, we know that $x$ can be either $-9$ or $9$. Therefore, $x = 9$ is a possible solution to this equation.

Option 4: $-|-x| = -9$

Now, let's consider the fourth option: $-|-x| = -9$. To solve this equation, we need to find the values of $x$ that make the negative of the absolute value of $-x$ equal to $-9$. Since the negative of the absolute value of $-x$ is $9$, we know that $x$ can be either $-9$ or $9$. Therefore, $x = 9$ is a possible solution to this equation.

Option 5: $|x| = -9$

Next, let's examine the fifth option: $|x| = -9$. To solve this equation, we need to find the values of $x$ that make the absolute value of $x$ equal to $-9$. However, the absolute value of a number cannot be negative, so there are no solutions to this equation. Therefore, $x = 9$ is not a possible solution to this equation.

Option 6: $|-x| = 9$

Now, let's consider the sixth option: $|-x| = 9$. To solve this equation, we need to find the values of $x$ that make the absolute value of $-x$ equal to $9$. Since the absolute value of $-x$ is $9$, we know that $x$ can be either $-9$ or $9$. Therefore, $x = 9$ is a possible solution to this equation.

Option 7: $|-x| = -9$

Finally, let's examine the seventh option: $|-x| = -9$. To solve this equation, we need to find the values of $x$ that make the absolute value of $-x$ equal to $-9$. However, the absolute value of a number cannot be negative, so there are no solutions to this equation. Therefore, $x = 9$ is not a possible solution to this equation.

Conclusion

In conclusion, we have examined each of the seven options and determined whether $x = 9$ is a possible solution. We found that $x = 9$ is a possible solution to the following equations:

• $|x| = 9$
• $-|x| = 9$
• $-|-x| = 9$
• $-|-x| = -9$
• $|-x| = 9$

However, $x = 9$ is not a possible solution to the following equations:

• $|x| = -9$
• $|-x| = -9$

We hope this article has provided a clear understanding of how to solve absolute value equations and has helped you to identify the possibilities of the equation $x = 9$ as a solution.

Introduction

In our previous article, we explored the possibilities of the equation $x = 9$ as a solution to various absolute value equations. We examined each option carefully and determined whether $x = 9$ is a possible solution. In this article, we will answer some of the most frequently asked questions about absolute value equations.

Q: What is the definition of absolute value?

A: The absolute value of a number is its distance from zero on the number line. In other words, it is the value of the number without considering its sign.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to find the values of the variable that make the absolute value expression equal to a certain value. You can do this by considering two cases: one where the expression inside the absolute value is positive, and one where it is negative.

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

A: The expression $|x|$ represents the absolute value of $x$, while the expression $|-x|$ represents the absolute value of $-x$. In other words, $|x|$ is the distance of $x$ from zero, while $|-x|$ is the distance of $-x$ from zero.

Q: Can the absolute value of a number be negative?

A: No, the absolute value of a number cannot be negative. The absolute value of a number is always non-negative, and it is equal to zero if and only if the number is zero.

Q: How do I determine whether an absolute value equation has a solution?

A: To determine whether an absolute value equation has a solution, you need to check whether the expression inside the absolute value is equal to the value on the other side of the equation. If it is, then the equation has a solution. If it is not, then the equation does not have a solution.

Q: Can an absolute value equation have more than one solution?

A: Yes, an absolute value equation can have more than one solution. For example, the equation $|x| = 9$ has two solutions: $x = 9$ and $x = -9$.

Q: How do I graph an absolute value function?

A: To graph an absolute value function, you need to graph the two cases: one where the expression inside the absolute value is positive, and one where it is negative. The graph of the absolute value function will be a V-shaped graph with its vertex at the origin.

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, the equation $|x - 5| = 3$ can be used to model the distance between two points on a number line.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about absolute value equations. We hope this article has provided a clear understanding of how to solve absolute value equations and has helped you to identify the possibilities of the equation $x = 9$ as a solution. If you have any further questions, please don't hesitate to ask.

Additional Resources

If you want to learn more about absolute value equations, we recommend the following resources:

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• Wolfram Alpha: Absolute Value Equations

We hope this article has been helpful in your understanding of absolute value equations.