What Is The Solution Set Of $-|-x|=-12$?A. No Solution B. $\{-12\}$ C. $\{12\}$ D. $\{-12, 12\}$

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Understanding the Absolute Value Equation

The given equation is −∣−x∣=−12-|-x|=-12. To solve this equation, we need to understand the concept of absolute value and how it affects the equation. 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.

Breaking Down the Equation

The equation −∣−x∣=−12-|-x|=-12 can be broken down into two parts:

  1. ∣−x∣=12|-x| = 12
  2. −∣−x∣=−12-|-x| = -12

Solving the First Part of the Equation

The first part of the equation is ∣−x∣=12|-x| = 12. Since the absolute value of a number is always non-negative, we can remove the absolute value sign and write the equation as −x=12-x = 12 or −x=−12-x = -12.

Solving the Second Part of the Equation

The second part of the equation is −∣−x∣=−12-|-x| = -12. Since the absolute value of a number is always non-negative, we can remove the absolute value sign and write the equation as −x=−12-x = -12.

Combining the Results

Now, let's combine the results from both parts of the equation. We have two possible equations:

  1. −x=12-x = 12
  2. −x=−12-x = -12

Solving for x

To solve for x, we need to isolate x in both equations.

  1. −x=12⇒x=−12-x = 12 \Rightarrow x = -12
  2. −x=−12⇒x=12-x = -12 \Rightarrow x = 12

Conclusion

The solution set of the equation −∣−x∣=−12-|-x|=-12 is {−12,12}\{-12, 12\}. This means that the equation has two solutions: x = -12 and x = 12.

Final Answer

The final answer is D. {−12,12}\{-12, 12\}.

Understanding the Concept of Absolute Value

The concept of absolute value is crucial in solving equations involving absolute value. 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.

Real-World Applications of Absolute Value

Absolute value has many real-world applications, such as:

  • Measuring distances: The absolute value of a number can be used to measure distances between two points on a number line.
  • Representing financial data: The absolute value of a number can be used to represent financial data, such as the absolute value of a stock's price change.
  • Modeling physical phenomena: The absolute value of a number can be used to model physical phenomena, such as the absolute value of a temperature change.

Common Mistakes to Avoid

When solving equations involving absolute value, there are several common mistakes to avoid:

  • Not considering the two possible cases: When solving an equation involving absolute value, it's essential to consider both possible cases: the case where the expression inside the absolute value is positive, and the case where the expression inside the absolute value is negative.
  • Not removing the absolute value sign: When solving an equation involving absolute value, it's essential to remove the absolute value sign to simplify the equation.
  • Not checking for extraneous solutions: When solving an equation involving absolute value, it's essential to check for extraneous solutions, which are solutions that do not satisfy the original equation.

Conclusion

In conclusion, the solution set of the equation −∣−x∣=−12-|-x|=-12 is {−12,12}\{-12, 12\}. This means that the equation has two solutions: x = -12 and x = 12. The concept of absolute value is crucial in solving equations involving absolute value, and there are several common mistakes to avoid when solving these types of equations.

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, without considering direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two possible cases: the case where the expression inside the absolute value is positive, and the case where the expression inside the absolute value is negative. You then solve each case separately and check for extraneous solutions.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that does not satisfy the original equation. When solving an absolute value equation, it's essential to check for extraneous solutions to ensure that the solution is valid.

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 check if it satisfies the equation. If the solution does not satisfy the equation, it is an extraneous solution.

Q: What is the difference between an absolute value equation and a linear equation?

A: An absolute value equation is an equation that involves the absolute value of a variable or expression, while a linear equation is an equation that involves a linear expression. Absolute value equations are more complex than linear equations and require a different approach to solve.

Q: Can I use algebraic methods to solve absolute value equations?

A: Yes, you can use algebraic methods to solve absolute value equations. However, you need to be careful when using algebraic methods, as they can lead to extraneous solutions.

Q: How do I graph an absolute value equation?

A: To graph an absolute value equation, you need to graph the two cases separately: the case where the expression inside the absolute value is positive, and the case where the expression inside the absolute value is negative. You then combine the two graphs to get the final graph.

Q: What is the significance of absolute value in real-world applications?

A: Absolute value has many real-world applications, such as measuring distances, representing financial data, and modeling physical phenomena. It is a fundamental concept in mathematics and has many practical uses.

Q: Can I use absolute value to solve quadratic equations?

A: Yes, you can use absolute value to solve quadratic equations. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I determine the number of solutions to an absolute value equation?

A: To determine the number of solutions to an absolute value equation, you need to consider the two possible cases: the case where the expression inside the absolute value is positive, and the case where the expression inside the absolute value is negative. If both cases have solutions, then the equation has two solutions. If only one case has a solution, then the equation has one solution. If neither case has a solution, then the equation has no solutions.

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

A: Yes, you can use absolute value to solve systems of equations. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to model real-world problems?

A: To use absolute value to model real-world problems, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

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 the two possible cases
  • Not removing the absolute value sign
  • Not checking for extraneous solutions
  • Not using algebraic methods correctly
  • Not graphing the equation correctly

Q: How do I choose the correct method to solve an absolute value equation?

A: To choose the correct method to solve an absolute value equation, you need to consider the type of equation and the variables involved. You then choose the method that is most suitable for the equation.

Q: Can I use absolute value to solve inequalities?

A: Yes, you can use absolute value to solve inequalities. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to solve word problems?

A: To use absolute value to solve word problems, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

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

A: Some real-world applications of absolute value include:

  • Measuring distances
  • Representing financial data
  • Modeling physical phenomena
  • Solving systems of equations
  • Solving inequalities
  • Solving word problems

Q: Can I use absolute value to solve optimization problems?

A: Yes, you can use absolute value to solve optimization problems. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to solve optimization problems?

A: To use absolute value to solve optimization problems, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

Q: What are some common applications of absolute value in science and engineering?

A: Some common applications of absolute value in science and engineering include:

  • Modeling physical phenomena
  • Representing financial data
  • Measuring distances
  • Solving systems of equations
  • Solving inequalities
  • Solving word problems

Q: Can I use absolute value to solve problems in computer science?

A: Yes, you can use absolute value to solve problems in computer science. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to solve problems in computer science?

A: To use absolute value to solve problems in computer science, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

Q: What are some common applications of absolute value in economics?

A: Some common applications of absolute value in economics include:

  • Representing financial data
  • Modeling economic phenomena
  • Solving systems of equations
  • Solving inequalities
  • Solving word problems

Q: Can I use absolute value to solve problems in finance?

A: Yes, you can use absolute value to solve problems in finance. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to solve problems in finance?

A: To use absolute value to solve problems in finance, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

Q: What are some common applications of absolute value in business?

A: Some common applications of absolute value in business include:

  • Representing financial data
  • Modeling business phenomena
  • Solving systems of equations
  • Solving inequalities
  • Solving word problems

Q: Can I use absolute value to solve problems in management?

A: Yes, you can use absolute value to solve problems in management. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to solve problems in management?

A: To use absolute value to solve problems in management, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

Q: What are some common applications of absolute value in social sciences?

A: Some common applications of absolute value in social sciences include:

  • Representing social data
  • Modeling social phenomena
  • Solving systems of equations
  • Solving inequalities
  • Solving word problems

Q: Can I use absolute value to solve problems in psychology?

A: Yes, you can use absolute value to solve problems in psychology. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to solve problems in psychology?

A: To use absolute value to solve problems in psychology, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

Q: What are some common applications of absolute value in education?

A: Some common applications of absolute value in education include:

  • Representing educational data
  • Modeling educational phenomena
  • Solving systems of equations
  • Solving inequalities
  • Solving word problems

Q: Can I use absolute value to solve problems in healthcare?

A: Yes, you can use absolute value to solve problems in healthcare. However, you need to be careful when using absolute value, as it can lead to extraneous solutions.

Q: How do I use absolute value to solve problems in healthcare?

A: To use absolute value to solve problems in healthcare, you need to identify the variables and constants in the problem and represent them using absolute value. You then use algebraic methods to solve the equation and find the solution.

Q: What are some common applications of absolute value in environmental science?

A: Some common applications of absolute value in environmental science include:

  • Representing environmental data
  • Modeling environmental phenomena
  • Solving systems of equations
  • Solving inequalities
  • Solving word problems

Q: Can I use absolute value to solve problems in geography?