Select The Correct Answer.Solve The Following Equation: ∣ X ∣ + 8 = 17 |x| + 8 = 17 ∣ X ∣ + 8 = 17 A. X = − 25 X = -25 X = − 25 Or X = − 9 X = -9 X = − 9 B. X = 25 X = 25 X = 25 Or X = − 25 X = -25 X = − 25 C. X = 25 X = 25 X = 25 Or X = − 9 X = -9 X = − 9 D. X = 9 X = 9 X = 9 Or X = − 9 X = -9 X = − 9

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Introduction

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 the equation ∣x∣+8=17|x| + 8 = 17 and explore the different possible solutions.

Understanding Absolute Value

Absolute value, denoted by the symbol ∣x∣|x|, is the distance of a number from zero on the number line. It is always non-negative, and it can be thought of as the magnitude of a number. For example, the absolute value of −3-3 is 33, and the absolute value of 55 is also 55.

Solving Absolute Value Equations

To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative.

Case 1: Expression Inside Absolute Value is Positive

If the expression inside the absolute value is positive, then the absolute value equation becomes:

x+8=17x + 8 = 17

Subtracting 88 from both sides gives us:

x=9x = 9

This is one possible solution to the equation.

Case 2: Expression Inside Absolute Value is Negative

If the expression inside the absolute value is negative, then the absolute value equation becomes:

−x+8=17-x + 8 = 17

Subtracting 88 from both sides gives us:

−x=9-x = 9

Multiplying both sides by −1-1 gives us:

x=−9x = -9

This is another possible solution to the equation.

Combining the Solutions

We have found two possible solutions to the equation: x=9x = 9 and x=−9x = -9. These solutions satisfy the original equation ∣x∣+8=17|x| + 8 = 17.

Conclusion

Solving absolute value equations requires considering two cases: one where the expression inside the absolute value is positive, and another where it is negative. By following these steps, we can find all possible solutions to the equation. In this article, we solved the equation ∣x∣+8=17|x| + 8 = 17 and found that the correct answer is x=9x = 9 or x=−9x = -9.

Final Answer

The final answer is:

  • D. x=9x = 9 or x=−9x = -9

This is the correct answer to the equation ∣x∣+8=17|x| + 8 = 17.

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Introduction

Absolute value equations can be a challenging topic for many students. In this article, we will address some of the most frequently asked questions about absolute value equations, providing clear explanations and examples to help you better understand this concept.

Q&A

Q: What is an absolute value equation?

A: An absolute value equation is an equation that contains the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, and it is always non-negative.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative. You can then solve each case separately and combine the solutions to find all possible answers.

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

A: An absolute value equation is an equation that contains the absolute value of a variable or expression, while a linear equation is an equation that can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants. Absolute value equations can be more challenging to solve than linear equations because they involve the absolute value function.

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

A: Yes, you can use algebraic manipulations to solve absolute value equations. For example, you can add or subtract the same value from both sides of the equation, or multiply both sides by the same non-zero value. However, you need to be careful when using these manipulations, as they can sometimes lead to extraneous solutions.

Q: How do I know if a solution to an absolute value equation is extraneous?

A: A solution to an absolute value equation is extraneous if it does not satisfy the original equation. To check if a solution is extraneous, you can substitute it back into the original equation and see if it is true. If it is not true, then the solution is extraneous.

Q: Can I use a calculator to solve absolute value equations?

A: Yes, you can use a calculator to solve absolute value equations. However, you need to be careful when using a calculator, as it may not always give you the correct answer. It's always a good idea to check your answer by substituting it back into the original equation.

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 both cases (positive and negative) when solving an absolute value equation.
  • Not checking if a solution is extraneous.
  • Not using algebraic manipulations carefully.
  • Not substituting solutions back into the original equation to check if they are true.

Conclusion

Absolute value equations can be a challenging topic, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article and avoiding common mistakes, you can solve absolute value equations with confidence.

Final Tips

  • Practice, practice, practice! The more you practice solving absolute value equations, the more comfortable you will become with the concept.
  • Use algebraic manipulations carefully to avoid extraneous solutions.
  • Check your solutions by substituting them back into the original equation.
  • Don't be afraid to ask for help if you get stuck.

By following these tips and practicing regularly, you can become a master of absolute value equations and tackle even the most challenging problems with confidence.