Let's analyze the equation [tex]\( |-x| = -10 \)[/tex] step-by-step to determine the solution set.
1. Understand the Absolute Value:
- The absolute value of any real number is always non-negative. That is, [tex]\( |a| \geq 0 \)[/tex] for any real number [tex]\( a \)[/tex].
2. Apply Absolute Value to the Given Equation:
- Notice that [tex]\( |-x| \)[/tex] represents the absolute value of [tex]\(-x\)[/tex]. Given that the absolute value of any number must be non-negative, [tex]\( |-x| \geq 0 \)[/tex].
3. Examine the Right Side of the Equation:
- The right side of the equation is [tex]\(-10\)[/tex], which is a negative number.
4. Compare the Values:
- Since [tex]\( |-x| \geq 0 \)[/tex] and [tex]\(-10\)[/tex] is negative, there is an inherent contradiction. The left side representing an absolute value cannot equal a negative number.
5. Conclusion:
- Because of the contradiction, it is impossible for [tex]\( |-x| \)[/tex] to equal [tex]\(-10\)[/tex]. Therefore, there are no values of [tex]\( x \)[/tex] that can satisfy the equation [tex]\( |-x| = -10 \)[/tex].
Thus, the solution set for the equation [tex]\( |-x| = -10 \)[/tex] is:
[tex]\[ \text{no solution} \][/tex]
