Misha Found That The Equation $-|2x-10|-1=2$ Had Two Possible Solutions: $x=3.5$ And $x=-6.5$. Which Explains Whether Or Not Her Solutions Are Correct?A. She Is Correct, Because Both Solutions Satisfy The Equation.B. She Is

Introduction

In mathematics, solving equations is a crucial skill that requires attention to detail and a thorough understanding of algebraic concepts. Misha, a math enthusiast, recently encountered an equation that had two possible solutions: $x=3.5$ and $x=-6.5$. To verify whether her solutions are correct, we need to substitute each value back into the original equation and check if it holds true. In this article, we will delve into the world of algebra and explore the process of verifying Misha's solutions.

Understanding the Equation

The given equation is $-|2x-10|-1=2$. To begin, let's simplify the equation by removing the absolute value signs. We know that the absolute value of a number is its distance from zero on the number line, and it is always non-negative. Therefore, we can rewrite the equation as:

$$-(2x-10)-1=2$$

Simplifying the Equation

Now, let's simplify the equation further by combining like terms. We can start by distributing the negative sign to the terms inside the parentheses:

$$-2x+10-1=2$$

Next, we can combine the constant terms:

$$-2x+9=2$$

Verifying Misha's Solutions

Now that we have simplified the equation, we can substitute each of Misha's solutions back into the equation to verify whether they are correct. Let's start with the first solution, $x=3.5$.

Substituting $x=3.5$ into the equation, we get:

$$-2(3.5)+9=2$$

Expanding the equation, we get:

$$-7+9=2$$

Simplifying further, we get:

$$2=2$$

As we can see, the equation holds true when $x=3.5$. Therefore, Misha's first solution is correct.

Verifying the Second Solution

Now, let's verify the second solution, $x=-6.5$. Substituting $x=-6.5$ into the equation, we get:

$$-2(-6.5)+9=2$$

Expanding the equation, we get:

$$13+9=2$$

Simplifying further, we get:

$$22=2$$

As we can see, the equation does not hold true when $x=-6.5$. Therefore, Misha's second solution is incorrect.

Conclusion

In conclusion, Misha's solutions to the equation $-|2x-10|-1=2$ are not both correct. While her first solution, $x=3.5$, is correct, her second solution, $x=-6.5$, is incorrect. This highlights the importance of verifying solutions to ensure that they satisfy the original equation.

Final Thoughts

Solving equations is a crucial skill in mathematics, and it requires attention to detail and a thorough understanding of algebraic concepts. By following the steps outlined in this article, we can verify whether Misha's solutions are correct and gain a deeper understanding of the equation. Whether you are a math enthusiast or a student, this article provides a valuable resource for learning and practicing algebraic concepts.

Additional Resources

For those who want to learn more about algebra and solving equations, here are some additional resources:

• Khan Academy: Algebra
• Mathway: Algebra Solver
• Wolfram Alpha: Algebra Calculator

By following these resources, you can gain a deeper understanding of algebraic concepts and improve your problem-solving skills.

FAQs

Q: What is the correct solution to the equation $-|2x-10|-1=2$? A: The correct solution to the equation is $x=3.5$.

Q: Why is Misha's second solution incorrect? A: Misha's second solution, $x=-6.5$, is incorrect because it does not satisfy the original equation.

Q: How can I verify solutions to an equation? A: To verify solutions, substitute the value back into the original equation and check if it holds true.

Introduction

In our previous article, we explored the process of verifying Misha's solutions to the equation $-|2x-10|-1=2$. We found that her first solution, $x=3.5$, was correct, while her second solution, $x=-6.5$, was incorrect. In this article, we will answer some frequently asked questions (FAQs) about verifying solutions to equations.

Q: What is the purpose of verifying solutions to equations?

A: The purpose of verifying solutions to equations is to ensure that the solutions satisfy the original equation. This is an important step in solving equations, as it helps to eliminate incorrect solutions and confirm the validity of the solution.

Q: How do I verify solutions to an equation?

A: To verify solutions, substitute the value back into the original equation and check if it holds true. This involves plugging the solution into the equation and simplifying to see if the equation is true.

Q: What are some common mistakes to avoid when verifying solutions?

A: Some common mistakes to avoid when verifying solutions include:

• Not simplifying the equation correctly
• Not checking if the solution satisfies the original equation
• Not considering all possible solutions

Q: Can I use a calculator to verify solutions?

A: Yes, you can use a calculator to verify solutions. However, it's always a good idea to double-check your work by hand to ensure that the solution is correct.

Q: How do I know if a solution is correct or incorrect?

A: A solution is correct if it satisfies the original equation. If the solution does not satisfy the equation, it is incorrect.

Q: What are some tips for verifying solutions to equations?

A: Some tips for verifying solutions to equations include:

• Read the equation carefully and understand what it's asking for
• Simplify the equation correctly
• Check if the solution satisfies the original equation
• Consider all possible solutions

Q: Can I use a graphing calculator to verify solutions?

A: Yes, you can use a graphing calculator to verify solutions. Graphing calculators can help you visualize the equation and see if the solution is correct.

Q: How do I know if a solution is an extraneous solution?

A: An extraneous solution is a solution that is not valid or is not a solution to the original equation. You can determine if a solution is extraneous by checking if it satisfies the original equation.

Q: What are some common types of extraneous solutions?

A: Some common types of extraneous solutions include:

• Solutions that are not in the domain of the equation
• Solutions that are not real numbers
• Solutions that are not in the range of the equation

Conclusion

In conclusion, verifying solutions to equations is an important step in solving equations. By following the tips and guidelines outlined in this article, you can ensure that your solutions are correct and valid. Remember to always read the equation carefully, simplify correctly, and check if the solution satisfies the original equation.

Additional Resources

For those who want to learn more about verifying solutions to equations, here are some additional resources:

• Khan Academy: Algebra
• Mathway: Algebra Solver
• Wolfram Alpha: Algebra Calculator

By following these resources, you can gain a deeper understanding of algebraic concepts and improve your problem-solving skills.

FAQs

Q: What is the difference between a solution and an extraneous solution? A: A solution is a value that satisfies the original equation, while an extraneous solution is a value that is not valid or is not a solution to the original equation.

Q: How do I know if a solution is a valid solution? A: A solution is a valid solution if it satisfies the original equation and is in the domain of the equation.

Q: Can I use a calculator to check if a solution is valid? A: Yes, you can use a calculator to check if a solution is valid. However, it's always a good idea to double-check your work by hand to ensure that the solution is correct.

Q: What are some common mistakes to avoid when verifying solutions? A: Some common mistakes to avoid when verifying solutions include not simplifying the equation correctly, not checking if the solution satisfies the original equation, and not considering all possible solutions.