Solve The Equation For { X $}$: ${ |x-25|=12 }$

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Introduction

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Absolute value equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of absolute value. 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 $|x-25|=12$ as a case study to demonstrate the step-by-step process of solving absolute value equations.

Understanding Absolute Value

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Absolute value is a mathematical operation that returns the distance of a number from zero on the number line. It is denoted by the symbol $|x|$. The absolute value of a number $x$ is defined as:

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

Solving Absolute Value Equations

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To solve an absolute value equation of the form $|x-a|=b$, we need to consider two cases:

1. Case 1: $x-a=b$
2. Case 2: $x-a=-b$

We will use these two cases to solve the given equation $|x-25|=12$.

Case 1: $x-25=12$

In this case, we add 25 to both sides of the equation to isolate $x$:

$$x-25+25=12+25$$

$$x=37$$

So, one possible solution to the equation $|x-25|=12$ is $x=37$.

Case 2: $x-25=-12$

In this case, we add 25 to both sides of the equation to isolate $x$:

$$x-25+25=-12+25$$

$$x=13$$

So, another possible solution to the equation $|x-25|=12$ is $x=13$.

Conclusion

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In this article, we have demonstrated the step-by-step process of solving absolute value equations of the form $|x-a|=b$. We used the given equation $|x-25|=12$ as a case study to illustrate the two cases that need to be considered when solving absolute value equations. By following these cases, we were able to find two possible solutions to the equation: $x=37$ and $x=13$. We hope that this article has provided a clear understanding of how to solve absolute value equations and has been helpful in your mathematical journey.

Frequently Asked Questions

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Q: What is an absolute value equation?

A: An absolute value equation is a mathematical equation that involves the absolute value of a variable or expression.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: $x-a=b$ and $x-a=-b$. You then solve each case separately to find the possible solutions to the equation.

Q: What is the difference between the two cases in solving absolute value equations?

A: The two cases in solving absolute value equations are based on whether the expression inside the absolute value is positive or negative. If the expression is positive, you add $a$ to both sides of the equation. If the expression is negative, you subtract $a$ from both sides of the equation.

Q: Can I have more examples of solving absolute value equations?

A: Yes, we can provide more examples of solving absolute value equations. Please let us know what type of examples you would like to see, and we will do our best to provide them.

Additional Resources

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If you are looking for more resources on solving absolute value equations, here are a few suggestions:

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• Purplemath: Absolute Value Equations

These resources provide a more in-depth explanation of solving absolute value equations and offer additional examples and practice problems to help you master this concept.

Final Thoughts

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Solving absolute value equations requires a clear understanding of the properties of absolute value and the ability to consider two cases when solving the equation. By following the step-by-step process outlined in this article, you can confidently solve absolute value equations and apply this concept to a wide range of mathematical problems. Remember to practice regularly and seek help when needed to become proficient in solving absolute value equations.

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Introduction

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In our previous article, we discussed the concept of absolute value equations and provided a step-by-step guide on how to solve them. However, we understand that there may be many questions and doubts that readers may have regarding absolute value equations. In this article, we will address some of the most frequently asked questions about absolute value equations and provide clear and concise answers to help you better understand this concept.

Q&A

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Q: What is the definition of an absolute value equation?

A: An absolute value equation is a mathematical equation that involves the absolute value of a variable or expression. It is denoted by the symbol $|x|$, where $x$ is the variable or expression inside the absolute value.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: $x-a=b$ and $x-a=-b$. You then solve each case separately to find the possible solutions to the equation.

Q: What is the difference between the two cases in solving absolute value equations?

A: The two cases in solving absolute value equations are based on whether the expression inside the absolute value is positive or negative. If the expression is positive, you add $a$ to both sides of the equation. If the expression is negative, you subtract $a$ from both sides of the equation.

Q: Can I have more examples of solving absolute value equations?

A: Yes, we can provide more examples of solving absolute value equations. Please let us know what type of examples you would like to see, and we will do our best to provide them.

Q: How do I know which case to use when solving an absolute value equation?

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, use the first case ($x-a=b$). If the expression is negative, use the second case ($x-a=-b$).

Q: Can I have a list of common absolute value equations?

A: Yes, here are some common absolute value equations:

• $|x-3|=4$
• $|x+2|=5$
• $|x-1|=3$
• $|x+4|=2$

Q: How do I graph an absolute value equation?

A: To graph an absolute value equation, you need to plot the two cases separately. For the first case ($x-a=b$), plot the line $y=x-a+b$. For the second case ($x-a=-b$), plot the line $y=x-a-b$.

Q: Can I have more resources on solving absolute value equations?

A: Yes, here are some additional resources on solving absolute value equations:

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• Purplemath: Absolute Value Equations

Conclusion

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We hope that this Q&A article has provided you with a better understanding of absolute value equations and has addressed some of the most frequently asked questions about this concept. Remember to practice regularly and seek help when needed to become proficient in solving absolute value equations. If you have any further questions or concerns, please don't hesitate to reach out to us.

Additional Resources

---

If you are looking for more resources on solving absolute value equations, here are a few suggestions:

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• Purplemath: Absolute Value Equations

These resources provide a more in-depth explanation of solving absolute value equations and offer additional examples and practice problems to help you master this concept.

Final Thoughts

---

Solving absolute value equations requires a clear understanding of the properties of absolute value and the ability to consider two cases when solving the equation. By following the step-by-step process outlined in this article and practicing regularly, you can confidently solve absolute value equations and apply this concept to a wide range of mathematical problems.