Hiroto Solved The Equation $6 - 4|2x - 8| = -10$ For One Solution. His Work Is Shown Below:$\[ \begin{align*} 6 - 4|2x - 8| &= -10 \\ -4|2x - 8| &= -16 \\ |2x - 8| &= 4 \\ 2x - 8 &= 4 \\ 2x &= 12 \\ x &= 6 \end{align*} \\]What Is The

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Introduction

In mathematics, solving equations is a fundamental concept that requires a deep understanding of various algebraic techniques. Hiroto's work on the equation 6−4∣2x−8∣=−106 - 4|2x - 8| = -10 is a great example of how to approach and solve absolute value equations. In this article, we will delve into the details of Hiroto's solution and provide a step-by-step analysis of the equation.

Understanding Absolute Value Equations

Absolute value equations involve absolute value expressions, which can be positive or negative. To solve these equations, we need to consider both cases: when the expression inside the absolute value is positive and when it is negative. The general form of an absolute value equation is ∣ax+b∣=c|ax + b| = c, where aa, bb, and cc are constants.

Hiroto's Solution

Hiroto's work on the equation 6−4∣2x−8∣=−106 - 4|2x - 8| = -10 is shown below:

{ \begin{align*} 6 - 4|2x - 8| &= -10 \\ -4|2x - 8| &= -16 \\ |2x - 8| &= 4 \\ 2x - 8 &= 4 \\ 2x &= 12 \\ x &= 6 \end{align*} \}

Let's break down each step of Hiroto's solution:

Step 1: Isolate the Absolute Value Expression

The first step is to isolate the absolute value expression on one side of the equation. In this case, we have:

6−4∣2x−8∣=−106 - 4|2x - 8| = -10

To isolate the absolute value expression, we can add 4∣2x−8∣4|2x - 8| to both sides of the equation:

6=−10+4∣2x−8∣6 = -10 + 4|2x - 8|

Step 2: Simplify the Equation

Next, we can simplify the equation by combining like terms:

16=4∣2x−8∣16 = 4|2x - 8|

Step 3: Divide Both Sides by 4

To get rid of the coefficient of the absolute value expression, we can divide both sides of the equation by 4:

4=∣2x−8∣4 = |2x - 8|

Step 4: Consider Both Cases

Since the absolute value expression can be positive or negative, we need to consider both cases:

Case 1: 2x−8=42x - 8 = 4

Case 2: 2x−8=−42x - 8 = -4

Step 5: Solve for x

Now, we can solve for x in both cases:

Case 1: 2x−8=42x - 8 = 4

2x=122x = 12

x=6x = 6

Case 2: 2x−8=−42x - 8 = -4

2x=42x = 4

x=2x = 2

Step 6: Check the Solutions

Finally, we need to check the solutions to make sure they satisfy the original equation. Let's plug in x = 6 and x = 2 into the original equation:

For x = 6:

6−4∣2(6)−8∣=6−4∣12−8∣=6−4∣4∣=6−16=−106 - 4|2(6) - 8| = 6 - 4|12 - 8| = 6 - 4|4| = 6 - 16 = -10

For x = 2:

6−4∣2(2)−8∣=6−4∣4−8∣=6−4∣−4∣=6−16=−106 - 4|2(2) - 8| = 6 - 4|4 - 8| = 6 - 4|-4| = 6 - 16 = -10

Both solutions satisfy the original equation, but we need to be careful when dealing with absolute value equations. In this case, Hiroto's solution is correct, but we need to be aware of the possibility of extraneous solutions.

Conclusion

In conclusion, Hiroto's solution to the equation 6−4∣2x−8∣=−106 - 4|2x - 8| = -10 is a great example of how to approach and solve absolute value equations. By following the steps outlined above, we can solve these equations and find the correct solutions. However, we need to be careful when dealing with absolute value equations and check the solutions to make sure they satisfy the original equation.

Common Mistakes to Avoid

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

  • Not considering both cases: When dealing with absolute value equations, we need to consider both cases: when the expression inside the absolute value is positive and when it is negative.
  • Not checking the solutions: We need to check the solutions to make sure they satisfy the original equation.
  • Not being aware of extraneous solutions: In some cases, we may find extraneous solutions that do not satisfy the original equation.

Tips and Tricks

Here are some tips and tricks to help you solve absolute value equations:

  • Use the properties of absolute value to simplify the equation.
  • Consider both cases: When dealing with absolute value equations, we need to consider both cases: when the expression inside the absolute value is positive and when it is negative.
  • Check the solutions: We need to check the solutions to make sure they satisfy the original equation.
  • Be aware of extraneous solutions: In some cases, we may find extraneous solutions that do not satisfy the original equation.

Real-World Applications

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

  • Physics: Absolute value equations are used to describe the motion of objects in physics.
  • Engineering: Absolute value equations are used to design and optimize systems in engineering.
  • Computer Science: Absolute value equations are used in computer science to solve problems related to data analysis and machine learning.

Final Thoughts

Introduction

In our previous article, we delved into the details of Hiroto's solution to the equation 6−4∣2x−8∣=−106 - 4|2x - 8| = -10. We broke down each step of his solution and provided a step-by-step analysis of the equation. In this article, we will continue to explore the topic of absolute value equations and answer some frequently asked questions.

Q&A: Absolute Value Equations

Q: What is an absolute value equation?

A: An absolute value equation is an equation that involves an absolute value expression. The absolute value of a number is its distance from zero on the number line, without considering direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider both cases: when the expression inside the absolute value is positive and when it is negative. You can use the properties of absolute value to simplify the equation and then solve for the variable.

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, not checking the solutions, and not being aware of extraneous solutions.

Q: How do I check the solutions to an absolute value equation?

A: To check the solutions to an absolute value equation, you need to plug the solutions back into the original equation and make sure they satisfy the equation.

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

A: Absolute value equations have many real-world applications, including physics, engineering, and computer science.

Q: Can you provide an example of an absolute value equation?

A: Here is an example of an absolute value equation:

∣2x−5∣=3|2x - 5| = 3

To solve this equation, you would need to consider both cases: when the expression inside the absolute value is positive and when it is negative.

Q: How do I simplify an absolute value expression?

A: To simplify an absolute value expression, you can use the properties of absolute value. For example, if you have the expression ∣2x−5∣|2x - 5|, you can simplify it by writing it as 2x−52x - 5 or −(2x−5)-(2x - 5).

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

A: An absolute value equation is an equation that involves an absolute value expression, while a linear equation is an equation that involves a linear expression. For example, the equation ∣2x−5∣=3|2x - 5| = 3 is an absolute value equation, while the equation 2x−5=32x - 5 = 3 is a linear equation.

Conclusion

In conclusion, solving absolute value equations requires a deep understanding of algebraic techniques and a careful approach to solving the equation. By following the steps outlined above and being aware of common mistakes to avoid, we can solve these equations and find the correct solutions. We hope this article has provided you with a better understanding of absolute value equations and how to solve them.

Additional Resources

If you are looking for additional resources to help you learn more about absolute value equations, here are a few suggestions:

  • Khan Academy: Khan Academy has a comprehensive video series on absolute value equations that covers the basics and beyond.
  • Mathway: Mathway is an online math problem solver that can help you solve absolute value equations and other types of math problems.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve absolute value equations and other types of math problems.

Final Thoughts

In conclusion, absolute value equations are an important topic in mathematics that require a deep understanding of algebraic techniques and a careful approach to solving the equation. By following the steps outlined above and being aware of common mistakes to avoid, we can solve these equations and find the correct solutions. We hope this article has provided you with a better understanding of absolute value equations and how to solve them.