To determine the number of solutions for the equation
[tex]\[ |4x + 12| = 0, \][/tex]
we need to understand the properties of the absolute value function. The absolute value of any expression [tex]\( |A| \)[/tex] is defined as follows:
[tex]\[ |A| =
\begin{cases}
A & \text{if } A \geq 0, \\
-A & \text{if } A < 0.
\end{cases}
\][/tex]
The absolute value of an expression is always non-negative (i.e., [tex]\( \geq 0 \)[/tex]). Therefore, the only way for [tex]\( |4x + 12| \)[/tex] to equal 0 is if the expression inside the absolute value is itself 0. This leads us to the equation:
[tex]\[ 4x + 12 = 0. \][/tex]
We solve this simple linear equation as follows:
1. Subtract 12 from both sides:
[tex]\[ 4x = -12. \][/tex]
2. Divide both sides by 4:
[tex]\[ x = \frac{-12}{4} = -3. \][/tex]
Thus, the equation [tex]\( 4x + 12 = 0 \)[/tex] has exactly one solution, which is [tex]\( x = -3 \)[/tex].
Now reconsider the original equation [tex]\( |4x + 12| = 0 \)[/tex]. Since the only way an absolute value can be 0 is if its argument is 0, and we've determined [tex]\( 4x + 12 = 0 \)[/tex] has one solution, we conclude:
[tex]\[ |4x + 12| = 0 \][/tex]
has exactly one solution.
Therefore, the correct answer is:
One solution
