What Is The Solution Set Of $|-x| = -10$?A. $\{10\}$ B. $\{-10\}$ C. $\{-10, 10\}$ D. No Solution

by ADMIN 101 views

Introduction

In mathematics, 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 this article, we will explore the solution set of the equation ∣−x∣=−10|-x| = -10.

What is Absolute Value?

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line. It is denoted by the symbol ∣x∣|x| and is defined as:

∣x∣={x,if x≥0−x,if x<0|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}

Solving Absolute Value Equations

To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is non-negative, and another where it is negative.

Case 1: Non-Negative Expression

If the expression inside the absolute value is non-negative, then the absolute value equation becomes:

∣x∣=x|x| = x

In this case, we can simply remove the absolute value sign and solve the equation:

x=−10x = -10

However, this equation has no solution, since the left-hand side is always non-negative, while the right-hand side is negative.

Case 2: Negative Expression

If the expression inside the absolute value is negative, then the absolute value equation becomes:

∣x∣=−x|x| = -x

In this case, we can multiply both sides of the equation by −1-1 to get:

x=10x = 10

However, this equation also has no solution, since the left-hand side is always negative, while the right-hand side is positive.

Conclusion

In conclusion, the equation ∣−x∣=−10|-x| = -10 has no solution. This is because the absolute value of a number is always non-negative, while the right-hand side of the equation is negative.

Solution Set

The solution set of the equation ∣−x∣=−10|-x| = -10 is the set of all values of xx that satisfy the equation. In this case, the solution set is empty, since there are no values of xx that satisfy the equation.

Answer

The correct answer is:

  • D. No solution

Final Thoughts

Introduction

In our previous article, we explored the solution set of the equation ∣−x∣=−10|-x| = -10. We saw that the absolute value of a number is always non-negative, and that the right-hand side of the equation is negative. Therefore, the equation has no 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 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: one where the expression inside the absolute value is non-negative, and another where it is negative. You can then use the definition of absolute value to simplify the equation and solve for the variable.

Q: What is the definition of absolute value?

A: The definition of absolute value is:

∣x∣={x,if x≥0−x,if x<0|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}

Q: How do I know which case to use?

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is non-negative, you use the first case. If the expression is negative, you use the second case.

Q: What if the expression inside the absolute value is zero?

A: If the expression inside the absolute value is zero, then the absolute value equation becomes:

∣x∣=0|x| = 0

In this case, the only solution is x=0x = 0.

Q: Can I have multiple solutions to an absolute value equation?

A: Yes, it is possible to have multiple solutions to an absolute value equation. For example, consider the equation ∣x∣=10|x| = 10. In this case, the solutions are x=10x = 10 and x=−10x = -10.

Q: How do I graph an absolute value equation?

A: To graph an absolute value equation, you need to graph the two cases separately. For the first case, you graph the equation y=xy = x. For the second case, you graph the equation y=−xy = -x. The graph of the absolute value equation is the union of these two graphs.

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 using the definition of absolute value correctly
  • Not checking the sign of the expression inside the absolute value
  • Not graphing the two cases separately

Conclusion

In this article, we provided a Q&A guide to help you understand absolute value equations and how to solve them. We hope that this guide has been helpful in clarifying any questions you may have had about absolute value equations.