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

Introduction

Absolute value equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will explore the concept of absolute value equations and determine for which equations $x = 9$ is a possible solution.

Understanding Absolute Value

Before we dive into the equations, let's first understand the concept of absolute value. The absolute value of a number $x$, denoted by $|x|$, is the distance of $x$ from zero on the number line. In other words, it is the magnitude of $x$ without considering its direction.

For example, if $x = 5$, then $|x| = 5$. Similarly, if $x = -5$, then $|x| = 5$. This is because the distance of $-5$ from zero is the same as the distance of $5$ from zero.

Equation 1: $|x| = 9$

Let's start with the first equation: $|x| = 9$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Since $|x| = 9$, we know that the distance of $x$ from zero is $9$. This means that $x$ can be either $9$ or $-9$, because both $9$ and $-9$ are $9$ units away from zero.

Therefore, the solutions to the equation $|x| = 9$ are $x = 9$ and $x = -9$.

Equation 2: $-|x| = 9$

Now, let's consider the second equation: $-|x| = 9$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Since $-|x| = 9$, we know that the negative of the distance of $x$ from zero is $9$. This means that the distance of $x$ from zero is $-9$, which is not possible, because distance cannot be negative.

Therefore, the equation $-|x| = 9$ has no solution.

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

Next, let's consider the third equation: $-|-x| = 9$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Since $-|-x| = 9$, we know that the negative of the distance of $-x$ from zero is $9$. This means that the distance of $-x$ from zero is $-9$, which is not possible, because distance cannot be negative.

However, we can rewrite the equation as $|-x| = -9$. Since the distance of $-x$ from zero is $|-x|$, we know that $|-x| = 9$. This means that the distance of $x$ from zero is $9$, which is possible.

Therefore, the solutions to the equation $-|-x| = 9$ are $x = -9$.

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

Now, let's consider the fourth equation: $-|-x| = -9$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Since $-|-x| = -9$, we know that the negative of the distance of $-x$ from zero is $-9$. This means that the distance of $-x$ from zero is $9$, which is possible.

Therefore, the solutions to the equation $-|-x| = -9$ are $x = 9$.

Equation 5: $|x| = -9$

Next, let's consider the fifth equation: $|x| = -9$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Since $|x| = -9$, we know that the distance of $x$ from zero is $-9$, which is not possible, because distance cannot be negative.

Therefore, the equation $|x| = -9$ has no solution.

Equation 6: $|-x| = 9$

Now, let's consider the sixth equation: $|-x| = 9$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Since $|-x| = 9$, we know that the distance of $-x$ from zero is $9$, which is possible.

This means that the distance of $x$ from zero is also $9$, because the distance of $-x$ from zero is the same as the distance of $x$ from zero.

Therefore, the solutions to the equation $|-x| = 9$ are $x = 9$ and $x = -9$.

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

Finally, let's consider the seventh equation: $|-x| = -9$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Since $|-x| = -9$, we know that the distance of $-x$ from zero is $-9$, which is not possible, because distance cannot be negative.

Therefore, the equation $|-x| = -9$ has no solution.

Conclusion

In conclusion, we have explored the concept of absolute value equations and determined for which equations $x = 9$ is a possible solution. We have found that $x = 9$ is a possible solution for the following equations:

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

We have also found that $x = 9$ is not a possible solution for the following equations:

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

Introduction

In our previous article, we explored the concept of absolute value equations and determined for which equations $x = 9$ is a possible solution. In this article, we will provide a Q&A guide to help you understand absolute value equations and how to solve them.

Q: What is an absolute value equation?

A: An absolute value equation is an 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.

Q: How do I solve an absolute value equation?

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

1. The expression inside the absolute value is positive.
2. The expression inside the absolute value is negative.

For each case, you need to solve the equation separately and then combine the solutions.

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

A: The expression $|x|$ represents the distance of $x$ from zero, while the expression $|-x|$ represents the distance of $-x$ from zero. Since the distance of $-x$ from zero is the same as the distance of $x$ from zero, we have $|x| = |-x|$.

Q: Can I simplify an absolute value equation by removing the absolute value sign?

A: No, you cannot simplify an absolute value equation by removing the absolute value sign. The absolute value sign is a mathematical operation that takes the distance of an expression from zero, and it cannot be removed.

Q: How do I handle negative numbers in absolute value equations?

A: When working with negative numbers in absolute value equations, you need to remember that the absolute value of a negative number is its distance from zero, which is always positive. Therefore, you can remove the negative sign from the expression inside the absolute value.

Q: Can I use the same solution for both $|x| = 9$ and $|-x| = 9$?

A: Yes, you can use the same solution for both $|x| = 9$ and $|-x| = 9$. Since $|x| = |-x|$, the solutions to both equations are the same.

Q: How do I know if an absolute value equation has a solution?

A: To determine if an absolute value equation has a solution, you need to check if the expression inside the absolute value is equal to the absolute value of the expression. If the expression inside the absolute value is equal to the absolute value of the expression, then the equation has a solution.

Q: Can I use absolute value equations to solve systems of equations?

A: Yes, you can use absolute value equations to solve systems of equations. By using absolute value equations, you can eliminate variables and solve for the remaining variables.

Q: Are absolute value equations used in real-world applications?

A: Yes, absolute value equations are used in various real-world applications, such as physics, engineering, and economics. They are used to model real-world problems, such as distance, speed, and time.

Conclusion

In conclusion, absolute value equations are a powerful tool for solving mathematical problems. By understanding how to solve absolute value equations, you can apply them to various real-world applications and solve complex problems. We hope that this Q&A guide has provided you with a comprehensive understanding of absolute value equations and how to solve them.