91Ó°ÊÓ

Subtract 1 from both sides to express the inequality in a standard form. \[\frac{1}{x-3} - 1< 0\]. This can be rewritten as \[\frac{1 - (x - 3)}{x - 3} < 0\] simplifying it further gives \[\frac{-x+4}{x - 3} < 0\].

Critical values are those values for which the expression equals zero or is undefined. In this case, the expression equals zero when the numerator equals to zero, therefore, -x + 4 = 0, solving for x gives x = 4. The expression is also undefined at x = 3, therefore, the critical values are x = 3, 4.

Since the inequality sign is 'less than' but not 'equal to', the critical values are not in the solution set. Now the critical values divide the real number line into three parts: (-∞, 3), (3, 4), and (4, ∞). Test each interval by choosing an arbitrary test value from each, and plug it into the simplified inequality. If the result is true, then the interval is part of the solution.\n\nFor (-∞, 3), choose x = 2, substituting into the inequality gives \(\frac{-2+4}{2-3}\) which is negative, so (-∞, 3) is part of the solution. \n\nFor (3, 4), choose x = 3.5, substituting into the inequality gives \(\frac{-3.5+4}{3.5-3}\) which is positive so this interval is not part of the solution.\n\nLastly for (4, ∞), choose x = 5, substituting into the inequality gives \(\frac{-5+4}{5-3}\) which is negative, so (4, ∞) is also part of the solution.