Misha Found That The Equation − ∣ 2 X − 10 ∣ − 1 = 2 -|2x-10|-1=2 − ∣2 X − 10∣ − 1 = 2 Had Two Possible Solutions: X = 3.5 X=3.5 X = 3.5 And X = − 6.5 X=-6.5 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

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Introduction

In mathematics, solving equations is a fundamental concept that requires careful analysis and verification of solutions. Misha, a student, has found two possible solutions to the equation −∣2x−10∣−1=2-|2x-10|-1=2. In this article, we will verify whether her solutions are correct or not.

Understanding the Equation

The given equation is −∣2x−10∣−1=2-|2x-10|-1=2. To verify Misha's solutions, we need to understand the equation and its components. The equation involves absolute value, which means that the expression inside the absolute value can be either positive or negative.

Breaking Down the Equation

Let's break down the equation into smaller parts to understand it better.

  • The expression inside the absolute value is 2x−102x-10.
  • The absolute value of 2x−102x-10 is denoted by −∣2x−10∣-|2x-10|.
  • The equation is −∣2x−10∣−1=2-|2x-10|-1=2.

Verifying Misha's Solutions

Misha has found two possible solutions: x=3.5x=3.5 and x=−6.5x=-6.5. To verify whether these solutions are correct, we need to substitute them into the original equation and check if they satisfy the equation.

Verifying x = 3.5

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

−∣2(3.5)−10∣−1=∣−7∣−1=6−1=5-|2(3.5)-10|-1=|-7|-1=6-1=5

Since 5≠25 \neq 2, the solution x=3.5x=3.5 does not satisfy the equation.

Verifying x = -6.5

Substituting x=−6.5x=-6.5 into the equation, we get:

−∣2(−6.5)−10∣−1=∣−13∣−1=12−1=11-|2(-6.5)-10|-1=|-13|-1=12-1=11

Since 11≠211 \neq 2, the solution x=−6.5x=-6.5 does not satisfy the equation.

Conclusion

In conclusion, Misha's solutions x=3.5x=3.5 and x=−6.5x=-6.5 do not satisfy the equation −∣2x−10∣−1=2-|2x-10|-1=2. Therefore, she is incorrect in her solutions.

Why Did Misha's Solutions Fail?

There could be several reasons why Misha's solutions failed. Here are a few possible reasons:

  • Insufficient algebraic manipulation: Misha may not have performed the necessary algebraic manipulations to simplify the equation and find the correct solutions.
  • Incorrect substitution: Misha may have substituted the wrong values into the equation, leading to incorrect solutions.
  • Lack of verification: Misha may not have verified her solutions by substituting them back into the original equation, leading to incorrect solutions.

Tips for Solving Equations

Here are some tips for solving equations:

  • Simplify the equation: Simplify the equation by performing algebraic manipulations to make it easier to solve.
  • Check for extraneous solutions: Check for extraneous solutions by substituting them back into the original equation.
  • Verify solutions: Verify solutions by substituting them back into the original equation to ensure that they satisfy the equation.

Conclusion

Introduction

Solving equations is a fundamental concept in mathematics that requires careful analysis and verification of solutions. In this article, we will answer some frequently asked questions (FAQs) about solving equations.

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It is a mathematical statement that contains an equal sign (=) and two expressions on either side of the equal sign.

Q: What is the difference between an equation and an expression?

A: An expression is a group of numbers, variables, and mathematical operations that are combined to form a value. An equation, on the other hand, is a statement that two expressions are equal.

Q: How do I solve an equation?

A: To solve an equation, you need to isolate the variable (the unknown value) on one side of the equation. You can do this by performing algebraic manipulations such as addition, subtraction, multiplication, and division.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check if my solution is correct?

A: To check if your solution is correct, you need to substitute the solution back into the original equation and verify that it satisfies the equation.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid or is not a solution to the equation. Extraneous solutions can occur when you make a mistake in your algebraic manipulations or when you substitute a value into the equation that is not a solution.

Q: How do I avoid extraneous solutions?

A: To avoid extraneous solutions, you need to be careful when performing algebraic manipulations and when substituting values into the equation. You should also check your solutions by substituting them back into the original equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by performing algebraic manipulations such as addition, subtraction, multiplication, and division.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula or factor the equation. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Conclusion

In conclusion, solving equations is a fundamental concept in mathematics that requires careful analysis and verification of solutions. By following the tips and techniques outlined in this article, you can solve equations with confidence and accuracy.

Additional Resources

For more information on solving equations, you can consult the following resources:

  • Math textbooks: Math textbooks provide a comprehensive overview of solving equations and other mathematical concepts.
  • Online resources: Online resources such as Khan Academy, Mathway, and Wolfram Alpha provide interactive lessons and practice problems to help you learn and practice solving equations.
  • Math tutors: Math tutors can provide one-on-one instruction and guidance to help you learn and practice solving equations.