Solve The Absolute Value Equation For $x$.$|x-3|-10=2$A) $x=15, X=-9$B) $x=15, X=9$

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Introduction to Absolute Value Equations

Absolute value equations are a type of algebraic equation that involves the absolute value of a variable or expression. In this article, we will focus on solving absolute value equations of the form $|x-a| = b$, where $a$ and $b$ are constants. We will use the given equation $|x-3|-10=2$ as an example to demonstrate the steps involved in solving absolute value equations.

Understanding the Given Equation

The given equation is $|x-3|-10=2$. To solve this equation, we need to isolate the absolute value expression on one side of the equation. We can do this by adding 10 to both sides of the equation, which gives us $|x-3|=12$.

Solving Absolute Value Equations

There are two cases to consider when solving absolute value equations:

Case 1: The expression inside the absolute value is non-negative

In this case, the absolute value expression can be written as $x-a$, where $a$ is a constant. We can then solve the equation $x-a=b$, where $b$ is the value of the absolute value expression.

Case 2: The expression inside the absolute value is negative

In this case, the absolute value expression can be written as $-x+a$, where $a$ is a constant. We can then solve the equation $-x+a=b$, where $b$ is the value of the absolute value expression.

Applying the Cases to the Given Equation

In the given equation $|x-3|=12$, we need to consider both cases.

Case 1: The expression inside the absolute value is non-negative

In this case, we can write the equation as $x-3=12$. Solving for $x$, we get $x=15$.

Case 2: The expression inside the absolute value is negative

In this case, we can write the equation as $-(x-3)=12$. Solving for $x$, we get $-x+3=12$, which simplifies to $-x=9$, and finally $x=-9$.

Conclusion

In conclusion, the solution to the absolute value equation $|x-3|-10=2$ is $x=15$ and $x=-9$. This demonstrates the importance of considering both cases when solving absolute value equations.

Final Answer

The final answer is: A)x=15,x=−9\boxed{A) x=15, x=-9}

Additional Tips and Tricks

  • When solving absolute value equations, it's essential to consider both cases: when the expression inside the absolute value is non-negative and when it's negative.
  • Use the correct signs when solving the equation in each case.
  • Check your solutions by plugging them back into the original equation.

Common Mistakes to Avoid

  • Failing to consider both cases when solving absolute value equations.
  • Not using the correct signs when solving the equation in each case.
  • Not checking solutions by plugging them back into the original equation.

Real-World Applications

Absolute value equations have numerous real-world applications, including:

  • Physics: When solving problems involving distance, speed, and time, absolute value equations are often used.
  • Engineering: In designing systems, engineers use absolute value equations to model and solve problems.
  • Computer Science: Absolute value equations are used in algorithms and data structures to solve problems efficiently.

Practice Problems

  1. Solve the absolute value equation $|x+2|=5$.
  2. Solve the absolute value equation $|x-4|=3$.
  3. Solve the absolute value equation $|2x-1|=6$.

Solutions to Practice Problems

  1. The solution to the absolute value equation $|x+2|=5$ is $x=7$ and $x=-12$.
  2. The solution to the absolute value equation $|x-4|=3$ is $x=7$ and $x=1$.
  3. The solution to the absolute value equation $|2x-1|=6$ is $x=4$ and $x=-\frac{5}{2}$.

Conclusion

In conclusion, solving absolute value equations requires careful consideration of both cases and the correct signs when solving the equation in each case. By following the steps outlined in this article, you can solve absolute value equations with confidence.

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Frequently Asked Questions

Q: What is an absolute value equation?

A: An absolute value equation is a type of algebraic equation that involves the absolute value of a variable or expression. It is written in the form $|x-a| = b$, where $a$ and $b$ are constants.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: when the expression inside the absolute value is non-negative and when it's negative. You can then solve the equation in each case by using the correct signs.

Q: What are the two cases to consider when solving absolute value equations?

A: The two cases to consider are:

  • Case 1: The expression inside the absolute value is non-negative.
  • Case 2: The expression inside the absolute value is negative.

Q: How do I determine 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, use Case 1. If the expression is negative, use Case 2.

Q: What is the difference between Case 1 and Case 2?

A: In Case 1, the absolute value expression can be written as $x-a$, where $a$ is a constant. In Case 2, the absolute value expression can be written as $-x+a$, where $a$ is a constant.

Q: How do I solve the equation in each case?

A: To solve the equation in each case, you need to use the correct signs. In Case 1, you can add $a$ to both sides of the equation. In Case 2, you can subtract $a$ from both sides of the equation.

Q: What are some common mistakes to avoid when solving absolute value equations?

A: Some common mistakes to avoid are:

  • Failing to consider both cases when solving absolute value equations.
  • Not using the correct signs when solving the equation in each case.
  • Not checking solutions by plugging them back into the original equation.

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

A: Absolute value equations have numerous real-world applications, including:

  • Physics: When solving problems involving distance, speed, and time, absolute value equations are often used.
  • Engineering: In designing systems, engineers use absolute value equations to model and solve problems.
  • Computer Science: Absolute value equations are used in algorithms and data structures to solve problems efficiently.

Q: How can I practice solving absolute value equations?

A: You can practice solving absolute value equations by working through examples and exercises. You can also try solving absolute value equations on your own and checking your solutions by plugging them back into the original equation.

Additional Resources

  • Khan Academy: Absolute Value Equations
  • Mathway: Absolute Value Equations
  • Wolfram Alpha: Absolute Value Equations

Conclusion

In conclusion, solving absolute value equations requires careful consideration of both cases and the correct signs when solving the equation in each case. By following the steps outlined in this article and practicing with examples and exercises, you can become proficient in solving absolute value equations.