Let's carefully analyze the given equation [tex]\( |-x| = -10 \)[/tex].
1. Understanding the Absolute Value Function: The absolute value function, denoted as [tex]\(|a|\)[/tex], always outputs non-negative results. In mathematical terms, [tex]\(|a| \geq 0\)[/tex] for any real number [tex]\(a\)[/tex].
2. Applying Absolute Value to [tex]\(-x\)[/tex]: In our equation, the expression is [tex]\(|-x|\)[/tex]. Since the absolute value of any number is always non-negative, we have that [tex]\(|-x| \geq 0\)[/tex]. This means that for any value of [tex]\(x\)[/tex], [tex]\( |-x| \)[/tex] will always be a non-negative number (greater than or equal to 0).
3. Comparing with [tex]\(-10\)[/tex]: The equation states [tex]\( |-x| = -10 \)[/tex]. Here, [tex]\(-10\)[/tex] is a negative number. Since the left side of the equation (which is [tex]\(|-x|\)[/tex]) is always non-negative, it can never be equal to a negative number like [tex]\(-10\)[/tex].
4. Conclusion: Therefore, the equation [tex]\( |-x| = -10 \)[/tex] has no possible solutions. The non-negativity property of absolute values ensures that no value of [tex]\(x\)[/tex] will satisfy this equation.
Thus, the solution set of [tex]\( |-x| = -10 \)[/tex] is:
[tex]\[ \text{no solution} \][/tex]
