Analyze The Solution Shown:1. $-|-x|=7$ : Given 2. $ |-x|=-7$ : Multiplication Property Of Equality 3. $ -x=7 $ Or $ -x=-7 $ : Definition Of Absolute Value 4. $ X=-7 $ Or $ X=7 $ : Multiplication

Understanding Absolute Value Equations

Absolute value equations are a type of mathematical equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, it is the magnitude of the number. Absolute value equations are often used to model real-world problems that involve distances, temperatures, and other quantities that can be positive or negative.

The Given Equation

The given equation is $-|-x|=7$. This equation involves the absolute value of the variable $x$, which is denoted by $|-x|$. The absolute value of $x$ is equal to $7$, and the negative sign in front of the absolute value indicates that the result is negative.

Step 1: Understanding the Absolute Value

The absolute value of a number is its distance from zero on the number line. In this case, the absolute value of $x$ is equal to $7$, which means that the distance between $x$ and zero is $7$ units.

Step 2: Applying the Definition of Absolute Value

The definition of absolute value states that $|-x| = x$ if $x \geq 0$ and $|-x| = -x$ if $x < 0$. In this case, we have $|-x| = 7$, which means that $x$ is either positive or negative.

Step 3: Multiplication Property of Equality

The multiplication property of equality states that if $a = b$, then $ac = bc$ for any number $c$. In this case, we have $|-x| = 7$, and we can multiply both sides of the equation by $-1$ to get $-|-x| = -7$.

Step 4: Solving for x

Now we have two possible equations: $-x = 7$ and $-x = -7$. We can solve each equation separately to find the values of $x$.

Solving the First Equation

The first equation is $-x = 7$. To solve for $x$, we can multiply both sides of the equation by $-1$ to get $x = -7$.

Solving the Second Equation

The second equation is $-x = -7$. To solve for $x$, we can multiply both sides of the equation by $-1$ to get $x = 7$.

Conclusion

In conclusion, the solution to the absolute value equation $-|-x|=7$ is $x = -7$ or $x = 7$. This means that the value of $x$ can be either $-7$ or $7$.

Understanding the Solution

The solution to the absolute value equation $-|-x|=7$ involves understanding the definition of absolute value and applying the multiplication property of equality. The absolute value of a number is its distance from zero on the number line, and the multiplication property of equality states that if $a = b$, then $ac = bc$ for any number $c$.

Real-World Applications

Absolute value equations have many real-world applications, such as modeling distances, temperatures, and other quantities that can be positive or negative. For example, a company may want to know the distance between two locations, and the absolute value equation can be used to model this distance.

Common Mistakes

When solving absolute value equations, it is common to make mistakes such as:

• Not understanding the definition of absolute value
• Not applying the multiplication property of equality correctly
• Not solving for both possible values of $x$

Tips and Tricks

When solving absolute value equations, here are some tips and tricks to keep in mind:

• Always understand the definition of absolute value
• Apply the multiplication property of equality correctly
• Solve for both possible values of $x$
• Check your work by plugging the values of $x$ back into the original equation

Conclusion

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Frequently Asked Questions

Q: What is an absolute value equation?

A: An absolute value equation is a type of mathematical equation that involves the absolute value of a variable or 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 understand the definition of absolute value and apply the multiplication property of equality. You also need to solve for both possible values of the variable.

Q: What is the definition of absolute value?

A: The definition of absolute value states that $|-x| = x$ if $x \geq 0$ and $|-x| = -x$ if $x < 0$.

Q: How do I apply the multiplication property of equality?

A: The multiplication property of equality states that if $a = b$, then $ac = bc$ for any number $c$. To apply this property, you need to multiply both sides of the equation by the same number.

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 understanding the definition of absolute value
• Not applying the multiplication property of equality correctly
• Not solving for both possible values of the variable

Q: How do I check my work when solving an absolute value equation?

A: To check your work, you need to plug the values of the variable back into the original equation and see if it is true.

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

A: Absolute value equations have many real-world applications, such as modeling distances, temperatures, and other quantities that can be positive or negative.

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

A: Yes, here is an example of an absolute value equation: $|-x| = 7$. This equation involves the absolute value of the variable $x$, which is equal to $7$.

Q: How do I solve the equation $|-x| = 7$?

A: To solve the equation $|-x| = 7$, you need to apply the definition of absolute value and the multiplication property of equality. You also need to solve for both possible values of the variable.

Q: What are the solutions to the equation $|-x| = 7$?

A: The solutions to the equation $|-x| = 7$ are $x = -7$ and $x = 7$.

Q: Can you explain the concept of absolute value in more detail?

A: The concept of absolute value refers to the distance of a number from zero on the number line, without considering direction. In other words, it is the magnitude of the number.

Q: How do I apply the concept of absolute value in real-world problems?

A: You can apply the concept of absolute value in real-world problems by using it to model distances, temperatures, and other quantities that can be positive or negative.

Q: What are some tips and tricks for solving absolute value equations?

A: Some tips and tricks for solving absolute value equations include:

• Always understand the definition of absolute value
• Apply the multiplication property of equality correctly
• Solve for both possible values of the variable
• Check your work by plugging the values of the variable back into the original equation

Conclusion

In conclusion, absolute value equations are a type of mathematical equation that involves the absolute value of a variable or expression. To solve an absolute value equation, you need to understand the definition of absolute value and apply the multiplication property of equality. You also need to solve for both possible values of the variable. By following these steps and avoiding common mistakes, you can successfully solve absolute value equations and apply the concept of absolute value in real-world problems.