Let's analyze the given equation [tex]\(-|-x| = -12\)[/tex].
An essential property of the absolute value function is that the absolute value of any number is always non-negative, meaning [tex]\(|-x| \geq 0\)[/tex] for all [tex]\(x\)[/tex].
Given the left-hand side of the equation is [tex]\(-|-x|\)[/tex]:
1. [tex]\(|-x|\)[/tex] represents the absolute value of [tex]\(-x\)[/tex], which is the same as the absolute value of [tex]\(x\)[/tex], as [tex]\(|-x| = |x|\)[/tex].
2. Since the absolute value of any number [tex]\(x\)[/tex] is always non-negative ([tex]\(|x| \geq 0\)[/tex]), multiplying it by -1 would make it non-positive ([tex]\(-|x| \leq 0\)[/tex]).
Therefore, [tex]\(-|x|\)[/tex] is always less than or equal to zero.
The equation thus can be written as:
[tex]\[
-|x| = -12
\][/tex]
Since [tex]\(-12\)[/tex] is a negative number, and [tex]\(-|x|\)[/tex] is always non-positive, there can be no value of [tex]\(x\)[/tex] that makes [tex]\(-|x| = -12\)[/tex] true.
Hence, the solution set for the equation [tex]\(-|-x| = -12\)[/tex] is:
[tex]\[
\text{No solution}
\][/tex]
