This is the amount the customer will pay after the discount and bonus

Oct 13, 2025 by talynt.org 84 views

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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 this article, we will explore the concept of equivalent relationships in absolute value equations, specifically focusing on the equation $|3x-9|-24=-12$ .

The Given Equation

The given equation is $|3x-9|-24=-12$ . To find equivalent relationships, we need to simplify this equation and express it in different forms.

Simplifying the Equation

To simplify the equation, we can start by adding 24 to both sides of the equation:

$|3x-9|-24+24=-12+24$

This simplifies to:

$|3x-9|=12$

Equivalent Relationships

Now that we have simplified the equation, we can express it in different forms to find equivalent relationships. Let's analyze each option:

Option A: $x=-9$ or $x=15$

To determine if this option is equivalent, we need to substitute the values of $x$ into the original equation and check if it holds true.

For $x=-9$ :

$|3(-9)-9|=|(-27)-9|=|-36|=36$

This is not equal to 12, so option A is not equivalent.

For $x=15$ :

$|3(15)-9|=|45-9|=|36|=36$

This is also not equal to 12, so option A is not equivalent.

Option B: $|3x-9|=12$

This option is equivalent to the simplified equation we obtained earlier. Therefore, option B is a correct answer.

Option C: $3x-9=-12$ or $3x-9=12$

To determine if this option is equivalent, we need to substitute the values of $3x-9$ into the original equation and check if it holds true.

For $3x-9=-12$ :

$|3x-9|-24=-12$

Substituting $3x-9=-12$ into the equation:

$|-12|-24=-12$

This is not equal to 12, so the first part of option C is not equivalent.

For $3x-9=12$ :

$|3x-9|-24=-12$

Substituting $3x-9=12$ into the equation:

$|12|-24=-12$

This is not equal to 12, so the second part of option C is not equivalent.

Option D: $|3x-9|=36$

This option is not equivalent to the simplified equation we obtained earlier. Therefore, option D is not a correct answer.

Option E: $3x-9=-36$

To determine if this option is equivalent, we need to substitute the value of $3x-9$ into the original equation and check if it holds true.

$|3x-9|-24=-12$

Substituting $3x-9=-36$ into the equation:

$|-36|-24=-12$

This is not equal to 12, so option E is not equivalent.

Conclusion

In conclusion, the correct answers are:

Option B: $|3x-9|=12$

This is the only option that is equivalent to the given equation $|3x-9|-24=-12$ . The other options are not equivalent, and therefore, are not correct answers.

Final Thoughts

Absolute value equations can be complex and challenging to work with. However, by simplifying the equation and expressing it in different forms, we can find equivalent relationships and determine the correct answers. This article has demonstrated how to find equivalent relationships in absolute value equations, specifically focusing on the equation $|3x-9|-24=-12$ .

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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 simplify an absolute value equation?

A: To simplify an absolute value equation, you can start by isolating the absolute value expression on one side of the equation. Then, you can use the definition of absolute value to rewrite the equation in a simpler form.

Q: What is the definition of absolute value?

A: The definition of absolute value is the distance of a number from zero on the number line, without considering direction. In other words, the absolute value of a number is its magnitude or size.

Q: How do I find equivalent relationships in an absolute value equation?

A: To find equivalent relationships in an absolute value equation, you can start by simplifying the equation and expressing it in different forms. You can also use the definition of absolute value to rewrite the equation in a simpler form.

Q: What are some common mistakes to avoid when working with absolute value equations?

A: Some common mistakes to avoid when working with absolute value equations include:

Not isolating the absolute value expression on one side of the equation
Not using the definition of absolute value to rewrite the equation in a simpler form
Not considering both positive and negative cases when working with absolute value equations

Q: How do I determine if two absolute value equations are equivalent?

A: To determine if two absolute value equations are equivalent, you can start by simplifying each equation and expressing it in a simpler form. You can then compare the two simplified equations to determine if they are equivalent.

Q: What is the importance of equivalent relationships in absolute value equations?

A: Equivalent relationships in absolute value equations are important because they allow us to simplify complex equations and express them in different forms. This can make it easier to solve the equation and find the solution.

Q: Can you provide an example of finding equivalent relationships in an absolute value equation?

A: Let's consider the equation $|3x-9|-24=-12$ . To find equivalent relationships, we can start by simplifying the equation and expressing it in different forms.

$|3x-9|-24=-12$

Adding 24 to both sides of the equation:

$|3x-9|=12$

This is an equivalent relationship to the original equation.

Q: How do I know if an absolute value equation has a solution?

A: To determine if an absolute value equation has a solution, you can start by simplifying the equation and expressing it in a simpler form. If the simplified equation has a solution, then the original equation also has a solution.

Q: Can you provide an example of an absolute value equation with no solution?

A: Let's consider the equation $|3x-9|=0$ . To determine if this equation has a solution, we can start by simplifying the equation and expressing it in a simpler form.

$|3x-9|=0$

This equation has no solution because the absolute value of any number is always greater than or equal to zero.

Q: How do I graph an absolute value equation?

A: To graph an absolute value equation, you can start by simplifying the equation and expressing it in a simpler form. You can then use the definition of absolute value to graph the equation on a number line.

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

A: Let's consider the equation $|3x-9|=12$ . To graph this equation, we can start by simplifying the equation and expressing it in a simpler form.

$|3x-9|=12$

This equation can be graphed on a number line by finding the points where the absolute value expression is equal to 12.

Q: How do I use absolute value equations in real-world applications?

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

Modeling distance and time relationships
Modeling financial transactions
Modeling physical phenomena

Q: Can you provide an example of using absolute value equations in a real-world application?

A: Let's consider a scenario where a company is offering a discount on a product. The discount is 10% off the original price, but the company is also offering a $5 bonus for purchasing the product. If the original price of the product is $100, how much will the customer pay after the discount and bonus?

Let's use an absolute value equation to model this scenario:

$|x-100|=10$

This equation represents the discount and bonus. To find the amount the customer will pay, we can simplify the equation and express it in a simpler form.

$|x-100|=10$

Subtracting 100 from both sides of the equation:

$|x-100-100|=10-100$

This simplifies to:

$|x-200|=-90$

This equation has no solution because the absolute value of any number is always greater than or equal to zero.

However, we can use the definition of absolute value to rewrite the equation in a simpler form.

$|x-200|=-90$

This equation can be rewritten as:

$x-200=-90$

Adding 200 to both sides of the equation:

$x=110$

This is the amount the customer will pay after the discount and bonus.

Q: How do I choose the correct method for solving an absolute value equation?

A: To choose the correct method for solving an absolute value equation, you can start by analyzing the equation and determining the best approach. Some common methods for solving absolute value equations include:

Simplifying the equation and expressing it in a simpler form
Using the definition of absolute value to rewrite the equation in a simpler form
Graphing the equation on a number line

Q: Can you provide an example of choosing the correct method for solving an absolute value equation?

A: Let's consider the equation $|3x-9|=12$ . To solve this equation, we can start by simplifying the equation and expressing it in a simpler form.

$|3x-9|=12$

This equation can be simplified by adding 9 to both sides of the equation:

$|3x|=21$

This equation can be rewritten as:

$3x=21$ or $3x=-21$

Solving for x:

$x=7$ or $x=-7$

This is the solution to the equation.

Q: How do I check my solution to an absolute value equation?

A: To check your solution to an absolute value equation, you can start by substituting the solution into the original equation and checking if it holds true.

Q: Can you provide an example of checking a solution to an absolute value equation?

A: Let's consider the equation $|3x-9|=12$ . To check the solution x=7, we can substitute x=7 into the original equation and check if it holds true.

$|3(7)-9|=|21-9|=|12|=12$

This equation holds true, so x=7 is a valid solution to the equation.

Q: How do I use technology to solve absolute value equations?

A: Technology can be used to solve absolute value equations in a variety of ways, including:

Using graphing calculators to graph the equation on a number line
Using computer algebra systems to simplify the equation and express it in a simpler form
Using online tools to solve the equation and find the solution

Q: Can you provide an example of using technology to solve an absolute value equation?

A: Let's consider the equation $|3x-9|=12$ . To solve this equation using technology, we can use a graphing calculator to graph the equation on a number line.

Using a graphing calculator, we can graph the equation and find the points where the absolute value expression is equal to 12.

The graph shows that the equation has two solutions: x=7 and x=-7.

This is the solution to the equation.

Q: How do I use absolute value equations in mathematical modeling?

A: Absolute value equations have many applications in mathematical modeling, including:

Modeling distance and time relationships
Modeling financial transactions
Modeling physical phenomena

Q: Can you provide an example of using absolute value equations in mathematical modeling?

Let's use an absolute value equation to model this scenario:

$|x-100|=10$

This equation represents the discount and bonus. To find the amount the customer will pay, we can simplify the equation and express it in a simpler form.

$|x-100|=10$

Subtracting 100 from both sides of the equation:

$|x-100-100|=10-100$

This simplifies to:

$|x-200|=-90$

This equation has no solution because the absolute value of any number is always greater than or equal to zero.

However, we can use the definition of absolute value to rewrite the equation in a simpler form.

$|x-200|=-90$

This equation can be rewritten as:

$x-200=-90$

Adding 200 to both sides of the equation:

$x=110$

This is the amount the customer will pay after the discount and bonus