Savanah Solved The Equation 3 + 4 ∣ X 2 + 3 ∣ = 11 3 + 4\left|\frac{x}{2} + 3\right| = 11 3 + 4 ​ 2 X ​ + 3 ​ = 11 For One Solution. Her Work Is Shown Below:1. $4\left|\frac{x}{2} + 3\right| = 8$2. $\left|\frac{x}{2} + 3\right| = 2$3. $\frac{x}{2} + 3 = 2$4.

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Introduction

Solving absolute value equations can be a challenging task, especially when dealing with complex expressions. In this article, we will explore the steps involved in solving absolute value equations, using the example of Savanah's work on the equation 3+4∣x2+3∣=113 + 4\left|\frac{x}{2} + 3\right| = 11. We will break down each step of her work and provide a detailed explanation of the process.

Step 1: Isolate the Absolute Value Expression

The first step in solving an absolute value equation is to isolate the absolute value expression. In this case, the absolute value expression is ∣x2+3∣\left|\frac{x}{2} + 3\right|. To isolate this expression, we need to subtract 3 from both sides of the equation and then divide by 4.

1. $3 + 4\left|\frac{x}{2} + 3\right| = 11$
2. $4\left|\frac{x}{2} + 3\right| = 8$

Step 2: Remove the Absolute Value Sign

Once we have isolated the absolute value expression, we can remove the absolute value sign by considering two cases: one where the expression inside the absolute value is positive, and one where it is negative.

2. $\left|\frac{x}{2} + 3\right| = 2$
3. $\frac{x}{2} + 3 = 2$ or $\frac{x}{2} + 3 = -2$

Step 3: Solve for x

Now that we have removed the absolute value sign, we can solve for x in each of the two cases.

3. $\frac{x}{2} + 3 = 2$
$\frac{x}{2} = -1$
$x = -2$

    

  1. x2+3=−2\frac{x}{2} + 3 = -2
x2=−5\frac{x}{2} = -5
x=−10x = -10

Discussion

Savanah's work on the equation 3+4∣x2+3∣=113 + 4\left|\frac{x}{2} + 3\right| = 11 is a great example of how to solve absolute value equations. By following the steps outlined above, we can isolate the absolute value expression, remove the absolute value sign, and solve for x in each of the two cases.

Conclusion

Solving absolute value equations requires a step-by-step approach. By isolating the absolute value expression, removing the absolute value sign, and solving for x in each of the two cases, we can find the solution to the equation. In this article, we have explored the steps involved in solving absolute value equations, using the example of Savanah's work on the equation 3+4∣x2+3∣=113 + 4\left|\frac{x}{2} + 3\right| = 11.

Common Mistakes to Avoid

When solving absolute value equations, there are several common mistakes to avoid. These include:

  • Not isolating the absolute value expression
  • Not removing the absolute value sign
  • Not solving for x in each of the two cases
  • Not considering the possibility of multiple solutions

Tips and Tricks

When solving absolute value equations, there are several tips and tricks to keep in mind. These include:

  • Always isolate the absolute value expression
  • Always remove the absolute value sign
  • Always solve for x in each of the two cases
  • Always consider the possibility of multiple solutions

Real-World Applications

Solving absolute value equations has many real-world applications. These include:

  • Physics: Solving absolute value equations is essential in physics, where it is used to model real-world phenomena such as motion and energy.
  • Engineering: Solving absolute value equations is essential in engineering, where it is used to design and optimize systems.
  • Computer Science: Solving absolute value equations is essential in computer science, where it is used to develop algorithms and solve problems.

Conclusion

Introduction

Solving absolute value equations can be a challenging task, but with the right guidance, anyone can master it. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving absolute value equations.

Q: What is an absolute value equation?

A: An absolute value equation is an equation that contains an absolute value expression, which is a value that is always non-negative.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to follow these steps:

  1. Isolate the absolute value expression
  2. Remove the absolute value sign
  3. Solve for x in each of the two cases

Q: What is the difference between a positive and negative absolute value?

A: When you remove the absolute value sign, you need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. This is because the absolute value of a negative number is always positive.

Q: How do I know which case to use?

A: To determine which case to use, you need to look at the expression inside the absolute value. If it is positive, you use the positive case. If it is negative, you use the negative case.

Q: What if I have a fraction inside the absolute value?

A: If you have a fraction inside the absolute value, you need to follow the same steps as before. However, you may need to simplify the fraction before removing the absolute value sign.

Q: Can I have multiple solutions to an absolute value equation?

A: Yes, you can have multiple solutions to an absolute value equation. This is because the absolute value expression can be positive or negative, resulting in two different solutions.

Q: How do I check my solutions?

A: To check your solutions, you need to plug them back into the original equation and verify that they are true.

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 isolating the absolute value expression
  • Not removing the absolute value sign
  • Not solving for x in each of the two cases
  • Not considering the possibility of multiple solutions

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

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

  • Physics: Solving absolute value equations is essential in physics, where it is used to model real-world phenomena such as motion and energy.
  • Engineering: Solving absolute value equations is essential in engineering, where it is used to design and optimize systems.
  • Computer Science: Solving absolute value equations is essential in computer science, where it is used to develop algorithms and solve problems.

Conclusion

Solving absolute value equations can be a challenging task, but with the right guidance, anyone can master it. By following the steps outlined in this Q&A guide, you can learn how to solve absolute value equations and apply them to real-world problems.

Additional Resources

For more information on solving absolute value equations, check out the following resources:

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

Practice Problems

Try solving the following absolute value equations:

  1. ∣x−3∣=2|x - 3| = 2
  2. ∣2x+1∣=5|2x + 1| = 5
  3. ∣x−2∣=3|x - 2| = 3

Answer Key

  1. x=1x = 1 or x=5x = 5
  2. x=−2x = -2 or x=3x = 3
  3. x=1x = 1 or x=5x = 5