What Are The Possible Values For $p$ In The Equation Below? ∣ P ∣ = 12 |p| = 12 ∣ P ∣ = 12 A. -12 B. -6 C. 0 D. 1 E. 6 F. 12 G. 24

Understanding the Absolute Value Equation

The given equation is $|p| = 12$, where $p$ is the variable. The absolute value of a number is its distance from zero on the number line, without considering direction. This means that the absolute value of any number is always non-negative.

Properties of Absolute Value

To solve the equation $|p| = 12$, we need to consider the properties of absolute value. The absolute value of a number $x$ can be defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

Solving the Equation

Using the definition of absolute value, we can rewrite the equation $|p| = 12$ as:

$$p = 12 \text{ or } p = -12$$

This is because the absolute value of $p$ is equal to $12$ if and only if $p$ is equal to $12$ or $-12$.

Possible Values for $p$

Based on the solution to the equation, the possible values for $p$ are:

• $p = 12$
• $p = -12$

Eliminating Incorrect Options

Let's examine the options provided:

A. -12 B. -6 C. 0 D. 1 E. 6 F. 12 G. 24

We can eliminate options B, C, D, E, and G because they do not satisfy the equation $|p| = 12$. Option B is too small, options C, D, and E are not equal to $12$ or $-12$, and option G is too large.

Conclusion

The possible values for $p$ in the equation $|p| = 12$ are $p = 12$ and $p = -12$. Therefore, the correct options are:

A. -12 F. 12

These two options satisfy the equation and are the only possible values for $p$.

Understanding the Absolute Value Equation

The given equation is $|p| = 12$, where $p$ is the variable. The absolute value of a number is its distance from zero on the number line, without considering direction. This means that the absolute value of any number is always non-negative.

Properties of Absolute Value

To solve the equation $|p| = 12$, we need to consider the properties of absolute value. The absolute value of a number $x$ can be defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

Solving the Equation

Using the definition of absolute value, we can rewrite the equation $|p| = 12$ as:

$$p = 12 \text{ or } p = -12$$

This is because the absolute value of $p$ is equal to $12$ if and only if $p$ is equal to $12$ or $-12$.

Possible Values for $p$

Based on the solution to the equation, the possible values for $p$ are:

• $p = 12$
• $p = -12$

Q&A

Q: What is the absolute value of a number?

A: The absolute value of a number is its distance from zero on the number line, without considering direction.

Q: How do you solve an absolute value equation?

A: To solve an absolute value equation, you need to consider the properties of absolute value and use the definition of absolute value to rewrite the equation.

Q: What are the possible values for $p$ in the equation $|p| = 12$?

A: The possible values for $p$ in the equation $|p| = 12$ are $p = 12$ and $p = -12$.

Q: Why are the absolute values of $12$ and $-12$ equal?

A: The absolute values of $12$ and $-12$ are equal because they are both $12$ units away from zero on the number line.

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

A: Yes, an example of an absolute value equation is $|x| = 5$. This equation can be rewritten as $x = 5$ or $x = -5$.

Q: How do you graph an absolute value function?

A: To graph an absolute value function, you need to graph the two cases separately: $y = x$ and $y = -x$. The graph of the absolute value function will be the combination of these two graphs.

Q: What are some common applications of absolute value?

A: Absolute value has many applications in mathematics, science, and engineering. Some common applications include:

• Modeling real-world phenomena, such as the distance between two points
• Solving optimization problems, such as finding the minimum or maximum value of a function
• Analyzing data, such as the spread of a dataset

Conclusion

The possible values for $p$ in the equation $|p| = 12$ are $p = 12$ and $p = -12$. We also discussed the properties of absolute value, how to solve absolute value equations, and some common applications of absolute value.