Does The Absolute Value Equation $|3x + 6| = -12$ Have A Solution?A. Yes, $x = \{-6, 2\}$ B. No, Because An Absolute Value (the Distance From Zero) Can Never Equal A Negative Number.

Does the Absolute Value Equation $|3x + 6| = -12$ Have a Solution?

Understanding Absolute Value Equations

Absolute value equations are a type of mathematical equation that involves the absolute value of an 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. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

The Equation $|3x + 6| = -12$

The given equation is $|3x + 6| = -12$. This equation involves the absolute value of the expression $3x + 6$, which is set equal to -12. To determine if this equation has a solution, we need to consider the properties of absolute value.

Properties of Absolute Value

One of the key properties of absolute value is that it is always non-negative. This means that the absolute value of any expression is always greater than or equal to zero. In other words, $|x| \geq 0$ for any real number $x$.

Applying the Properties of Absolute Value to the Equation

Now, let's apply this property to the given equation. We have $|3x + 6| = -12$. Since the absolute value of any expression is always non-negative, we know that $|3x + 6| \geq 0$. However, the equation states that $|3x + 6| = -12$, which is a negative number. This is a contradiction, as the absolute value of any expression cannot be negative.

Conclusion

Based on the properties of absolute value, we can conclude that the equation $|3x + 6| = -12$ has no solution. This is because the absolute value of any expression is always non-negative, and the equation states that the absolute value of $3x + 6$ is equal to a negative number.

Why the Equation Has No Solution

The equation $|3x + 6| = -12$ has no solution because it is based on a false assumption. The assumption is that the absolute value of $3x + 6$ can be equal to a negative number. However, as we have seen, the absolute value of any expression is always non-negative. Therefore, the equation $|3x + 6| = -12$ is a contradiction, and it has no solution.

Comparison with Other Options

Let's compare our conclusion with the other options. Option A states that the equation has a solution, and it provides two possible values for $x$. However, as we have seen, the equation has no solution because it is based on a false assumption. Option B states that the equation has no solution because an absolute value can never equal a negative number. This is consistent with our conclusion, and it is the correct answer.

Conclusion

In conclusion, the equation $|3x + 6| = -12$ has no solution. This is because the absolute value of any expression is always non-negative, and the equation states that the absolute value of $3x + 6$ is equal to a negative number. Therefore, the correct answer is option B, which states that the equation has no solution because an absolute value can never equal a negative number.

Final Answer

The final answer is B.
Frequently Asked Questions About Absolute Value Equations

Q: What is an absolute value equation?

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

Q: What is the property of absolute value that is relevant to solving absolute value equations?

A: One of the key properties of absolute value is that it is always non-negative. This means that the absolute value of any expression is always greater than or equal to zero.

Q: How does this property affect the solution of absolute value equations?

A: Since the absolute value of any expression is always non-negative, we know that the absolute value of an expression cannot be equal to a negative number. Therefore, if an absolute value equation states that the absolute value of an expression is equal to a negative number, it has no solution.

Q: Can you give an example of an absolute value equation that has no solution?

A: Yes, the equation $|3x + 6| = -12$ is an example of an absolute value equation that has no solution. This is because the absolute value of $3x + 6$ is always non-negative, and the equation states that it is equal to a negative number.

Q: How do you determine if an absolute value equation has a solution?

A: To determine if an absolute value equation has a solution, you need to check if the equation is consistent with the properties of absolute value. If the equation states that the absolute value of an expression is equal to a negative number, it has no solution.

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

A: An absolute value equation is a type of mathematical equation that involves the absolute value of an expression, while a quadratic equation is a type of mathematical equation that involves a quadratic expression. The key difference between the two is that absolute value equations involve the absolute value of an expression, while quadratic equations involve a quadratic expression.

Q: Can you give an example of a quadratic equation?

A: Yes, the equation $x^2 + 4x + 4 = 0$ is an example of a quadratic equation. This equation involves a quadratic expression, and it can be solved using the quadratic formula.

Q: How do you solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula. The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: Can you give an example of how to use the quadratic formula?

A: Yes, let's consider the quadratic equation $x^2 + 4x + 4 = 0$. To solve this equation, we need to use the quadratic formula. The coefficients of the equation are $a = 1$, $b = 4$, and $c = 4$. Plugging these values into the quadratic formula, we get:

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}$

Simplifying this expression, we get:

$x = \frac{-4 \pm \sqrt{16 - 16}}{2}$

$x = \frac{-4 \pm \sqrt{0}}{2}$

$x = \frac{-4}{2}$

$x = -2$

Therefore, the solution to the equation $x^2 + 4x + 4 = 0$ is $x = -2$.

Conclusion

In conclusion, absolute value equations are a type of mathematical equation that involves the absolute value of an expression. The key property of absolute value is that it is always non-negative, and this property affects the solution of absolute value equations. If an absolute value equation states that the absolute value of an expression is equal to a negative number, it has no solution.