Observe that \(\frac{x+ 5}{x-5}\) is zero at x = -5 and not defined at x = 5
Hence plotting these two points on number line
Now for x > 5, \(\frac{x+5}{x-5}\) is positive
For every root and not defined value of \(\frac{x+5}{x-5}\) the sign will change
We want greater than zero that is the positive part hence x < -5 and x > 5
x < -5 means x is from negative infinity to -5 and x > 5 means x is from 5 to infinity
Hence x ∈ (-∞, -5) U (5, ∞)
Hence solution of \(\frac{x}{x-5} > \frac{1}{2}\) is x ∈ (-∞, 2) U (5, ∞)
