Let's analyze each equation step-by-step to determine which ones have no solution:
1. Equation: [tex]\(-|x| = 0\)[/tex]
- The left side is the negative absolute value of [tex]\(x\)[/tex], which can be zero only if [tex]\(x = 0\)[/tex].
- Therefore, this equation has a solution at [tex]\( x = 0 \)[/tex].
2. Equation: [tex]\(|x| = -15\)[/tex]
- The absolute value of any number is always non-negative.
- There is no number [tex]\(x\)[/tex] whose absolute value can be negative.
- Therefore, this equation has no solution.
3. Equation: [tex]\(-|x| = 12\)[/tex]
- The left side is the negative absolute value of [tex]\(x\)[/tex], which is always non-positive.
- There is no number [tex]\(x\)[/tex] that would make [tex]\(-|x|\)[/tex] equal to a positive number like 12.
- Therefore, this equation has no solution.
4. Equation: [tex]\(-|-x| = 9\)[/tex]
- The left side is the negative absolute value of [tex]\(-x\)[/tex]. Since [tex]\(|-x| = |x|\)[/tex], [tex]\(-|-x|\)[/tex] is always non-positive.
- There is no number [tex]\(x\)[/tex] that would make [tex]\(-|x|\)[/tex] equal to a positive number like 9.
- Therefore, this equation has no solution.
5. Equation: [tex]\(-|-x| = -2\)[/tex]
- The left side is the negative absolute value of [tex]\(-x\)[/tex], which can be negative. In this case, when [tex]\(|-x| = |x| = 2\)[/tex], [tex]\(-|x|\)[/tex] would be [tex]\(-2\)[/tex].
- Therefore, this equation has a solution when [tex]\(x = 2\)[/tex] or [tex]\(x = -2\)[/tex].
From our analysis, the equations that have no solutions are:
- [tex]\(|x| = -15\)[/tex]
- [tex]\(-|x| = 12\)[/tex]
- [tex]\(-|-x| = 9\)[/tex]
Thus, the equations with no solutions are:
- [tex]\(2. \quad |x| = -15\)[/tex]
- [tex]\(3. \quad -|x| = 12\)[/tex]
- [tex]\(4. \quad -|-x| = 9\)[/tex]
These correspond to equations numbered 2, 3, and 4.
