Let's analyze each equation step by step to determine if it has any solutions or not:
1. Equation: [tex]\( -|x| = 0 \)[/tex]
- The absolute value function [tex]\( |x| \)[/tex] always yields a non-negative result. Taking the negative of a non-negative number, the only way for it to be zero is if the absolute value itself is zero.
- [tex]\( |x| = 0 \)[/tex] implies [tex]\( x = 0 \)[/tex].
- Therefore, this equation has exactly one solution: [tex]\( x = 0 \)[/tex].
- This equation has a solution.
2. Equation: [tex]\( |x| = -15 \)[/tex]
- The absolute value function [tex]\( |x| \)[/tex] always yields a non-negative result. It can never be negative.
- Since [tex]\(|x|\)[/tex] cannot equal -15, there are no values of [tex]\( x \)[/tex] that satisfy this equation.
- Therefore, this equation has no solution.
3. Equation: [tex]\( -|x| = 12 \)[/tex]
- The absolute value function [tex]\( |x| \)[/tex] always yields a non-negative result. Taking the negative of a non-negative number will always result in a non-positive number (i.e., [tex]\( \leq 0 \)[/tex]).
- Since [tex]\(-|x|\)[/tex] cannot be a positive number like 12, there are no values of [tex]\( x \)[/tex] that satisfy this equation.
- Therefore, this equation has no solution.
4. Equation: [tex]\( -|-x| = 9 \)[/tex]
- The expression [tex]\( |-x| \)[/tex] is the same as [tex]\( |x| \)[/tex], since the absolute value of a number is always non-negative.
- Therefore, [tex]\( -|-x| \)[/tex] is the same as [tex]\( -|x| \)[/tex], which must be non-positive (i.e., [tex]\( \leq 0 \)[/tex]).
- Since [tex]\(-|x|\)[/tex] cannot be a positive number like 9, there are no values of [tex]\( x \)[/tex] that satisfy this equation.
- Therefore, this equation has no solution.
5. Equation: [tex]\( -|-x| = -2 \)[/tex]
- The expression [tex]\( |-x| \)[/tex] is the same as [tex]\( |x| \)[/tex], since the absolute value of a number is always non-negative.
- Thus, rewriting the equation gives [tex]\( -|x| = -2 \)[/tex].
- For this to hold true, [tex]\( |x| \)[/tex] must equal 2. Hence, [tex]\( x \)[/tex] could be either 2 or -2.
- Therefore, this equation has two solutions: [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
- This equation has solutions.
Thus, the equations that have no solution are:
- [tex]\( |x| = -15 \)[/tex]
- [tex]\( -|x| = 12 \)[/tex]
- [tex]\( -|-x| = 9 \)[/tex]
