\(\frac{|{\text{x}}-2|}{({\text{x}}-2)}\) < 0 means we have to find values of x for which \(\frac{|{\text{x}}-2|}{({\text{x}}-2)}\) is negative Observe that the numerator |x – 2| is always positive because of mod, hence for \(\frac{|{\text{x}}-2|}{({\text{x}}-2)}\) to be a negative quantity the denominator (x - 2) has to be negative
That is x - 2 should be less than 0
⇒ x – 2 < 0
⇒ x < 2
Hence x should be less than 2 for\(\frac{|{\text{x}}-2|}{({\text{x}}-2)}\) < 0
x < 2 means x can take values from -∞ to 2 hence x ∈ (-∞, 2)
Hence the solution set for \(\frac{|{\text{x}}-2|}{({\text{x}}-2)}\) < 0 is (-∞, 2)
