Johan Found That The Equation − 2 ∣ 8 − X ∣ − 6 = − 12 -2|8-x|-6=-12 − 2∣8 − X ∣ − 6 = − 12 Had Two Possible Solutions: X = 5 X=5 X = 5 And X = − 11 X=-11 X = − 11 . Which Explains Whether His Solutions Are Correct?A. He Is Correct Because Both Solutions Satisfy The Equation.B. He Is Not Correct

Introduction

In mathematics, absolute value equations are a type of equation that involve absolute value expressions. These equations can be challenging to solve, but with the right approach, we can find the solutions. In this article, we will explore an absolute value equation and verify the solutions found by Johan.

The Equation

The equation given by Johan is $-2|8-x|-6=-12$. To verify his solutions, we need to substitute each solution into the equation and check if it holds true.

Substituting the Solutions

Let's start by substituting $x=5$ into the equation:

$-2|8-5|-6=-12$

$-2|3|-6=-12$

$-2(3)-6=-12$

$-6-6=-12$

$-12=-12$

Since the equation holds true for $x=5$, we can conclude that this solution is correct.

Next, let's substitute $x=-11$ into the equation:

$-2|8-(-11)|-6=-12$

$-2|19|-6=-12$

$-2(19)-6=-12$

$-38-6=-12$

$-44=-12$

Since the equation does not hold true for $x=-11$, we can conclude that this solution is incorrect.

Conclusion

In conclusion, Johan's solutions to the equation $-2|8-x|-6=-12$ are correct for $x=5$ and incorrect for $x=-11$. This verifies that the solution $x=5$ satisfies the equation, while the solution $x=-11$ does not.

Understanding Absolute Value Equations

Absolute value equations involve absolute value expressions, which can be challenging to solve. However, with the right approach, we can find the solutions. Here are some key concepts to understand when working with absolute value equations:

• Absolute value: The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$.
• Absolute value expression: An absolute value expression is an expression that involves an absolute value. For example, $|x-2|$ is an absolute value expression.
• Solving absolute value equations: To solve an absolute value equation, we need to isolate the absolute value expression and then solve for the variable.

Tips for Solving Absolute Value Equations

Here are some tips for solving absolute value equations:

• Isolate the absolute value expression: The first step in solving an absolute value equation is to isolate the absolute value expression.
• Solve for the variable: Once the absolute value expression is isolated, we can solve for the variable.
• Check the solutions: It's essential to check the solutions to ensure that they satisfy the equation.

Common Mistakes to Avoid

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

• Not isolating the absolute value expression: Failing to isolate the absolute value expression can make it challenging to solve the equation.
• Not checking the solutions: Failing to check the solutions can lead to incorrect answers.
• Not considering multiple solutions: Absolute value equations can have multiple solutions, so it's essential to consider all possible solutions.

Real-World Applications

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

• Physics: Absolute value equations are used to model real-world phenomena, such as the motion of objects.
• Engineering: Absolute value equations are used to design and optimize systems, such as electrical circuits.
• Computer Science: Absolute value equations are used in computer algorithms, such as sorting and searching.

Conclusion

Q: What is an absolute value equation?

A: An absolute value equation is a type of equation that involves an absolute value expression. The absolute value of a number is its distance from zero on the number line.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to isolate the absolute value expression and then solve for the variable. You can do this by using the following steps:

1. Isolate the absolute value expression.
2. Set up two equations: one with the positive value and one with the negative value.
3. Solve each equation separately.
4. Check the solutions to ensure that they satisfy the original equation.

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

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

Q: Can an absolute value equation have multiple solutions?

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

Q: How do I check if a solution is correct?

A: To check if a solution is correct, you need to substitute the solution into the original equation and check if it holds true. If the equation holds true, then the solution is correct.

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 checking the solutions.
• Not considering multiple solutions.

Q: Can absolute value equations be used in real-world applications?

A: Yes, absolute value equations can be used in real-world applications, such as physics, engineering, and computer science.

Q: How do I graph an absolute value equation?

A: To graph an absolute value equation, you need to graph the absolute value expression and then reflect it across the x-axis. This will give you the graph of the absolute value equation.

Q: What is the vertex form of an absolute value equation?

A: The vertex form of an absolute value equation is $y=a|x-h|+k$, where $(h,k)$ is the vertex of the graph.

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

A: Yes, you can use a calculator to solve absolute value equations. However, it's always a good idea to check the solutions by hand to ensure that they are correct.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to isolate the absolute value expression and then solve for the variable. You can do this by using the following steps:

1. Isolate the absolute value expression.
2. Set up two inequalities: one with the positive value and one with the negative value.
3. Solve each inequality separately.
4. Check the solutions to ensure that they satisfy the original inequality.

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

A: An absolute value inequality involves an absolute value expression and a comparison operator (such as <, >, ≤, or ≥), while an absolute value equation involves an absolute value expression and an equal sign (=).