What is the solution set of |-x| = - 10?

|a| = b gives

a = b or a = - b

so,

|-x| = - 10

gives

-x = - 10 or - x = 10

x = 10, - 10

now let us verify,

when x = 10, |-10| = + 10 and it is not = - 10

so, x = 10 is NOT a solution.

when x = - 10, | - (-10) | = |10| = 10 and it is not = - 10

so, x = - 10 is NOT a solution.

hence, this equation does not have a solution.

If we know that | ... | can never be negative, we can directly deduce that this equation does not have any solution.