Let's analyze each equation step-by-step to determine which ones have no solutions.
1. [tex]$-|x|=0$[/tex]
[tex]\[
|x| \text{ is the absolute value of } x, \text{ which is always non-negative.}
\][/tex]
If [tex]$-|x|=0$[/tex], then [tex]$x$[/tex] must be [tex]$0$[/tex] because:
[tex]\[
-|0| = -0 = 0
\][/tex]
Equation [tex]\(1\)[/tex] does have a solution: [tex]\(x = 0\)[/tex].
2. [tex]$|x|=-15$[/tex]
[tex]\[
|x| \text{ represents the absolute value of } x, \text{ which is always non-negative.}
\][/tex]
Thus, [tex]$|x|$[/tex] cannot be equal to a negative value. Therefore:
[tex]\[
|x| = -15 \quad \text{has no solution.}
\][/tex]
3. [tex]$-|x|=12$[/tex]
[tex]\[
-|x| \text{ indicates the negation of the absolute value of } x.
\][/tex]
Since [tex]$|x|$[/tex] is always non-negative, [tex]$-|x|$[/tex] is always non-positive. This means it can never be equal to a positive number:
[tex]\[
-|x| = 12 \quad \text{has no solution.}
\][/tex]
4. [tex]$-|-x|=9$[/tex]
[tex]\[
|-x| \text{ is equal to } |x|, \text{ since the absolute value of a number is always positive regardless of its sign.}
\][/tex]
Thus, [tex]$-|x|$[/tex] is non-positive and cannot be equal to a positive number:
[tex]\[
-|-x| = 9 \quad \text{has no solution.}
\][/tex]
5. [tex]$-|-x|=-2$[/tex]
[tex]\[
-|-x| \text{ equals } -|x| \text{, which is always non-positive.}
\][/tex]
Although [tex]$-|-x|$[/tex] can indeed be equal to a non-positive number, it would mean [tex]$|x| = 2$[/tex]. Thus:
\]
\quad -|-x| = -|x| = -2.
[tex]\[
\quad \text{has a solution in this case.}
So, the equations that have no solution are:
\[
\boxed{2, 3, 4}
\][/tex]
