Q.Solve the following inequality and write the solution set using interval notation: $\dfrac{2x}{x-4} \le 5$.

Since we don't know the sign of $x-4$, we cannot multiply across without first combining into one fraction. Subtract 5 from both sides: $\frac{2x}{x-4} - 5 \le 0$. Combine over a common denominator: $\frac{2x - 5(x-4)}{x-4} \le 0$, i.e. $\frac{2x-5x+20}{x-4} \le 0$, i.e. $\frac{-3x+20}{x-4} \le 0$. Multiply both sides by $-1$ (flip the inequality): $\frac{3x-20}{x-4} \ge 0$. The critical points are $x=\frac{20}{3}\approx 6.67$ (numerator zero) and $x=4$ (denominator zero, so x=4 must be EXCLUDED from the solution set no matter what). Testing regions: for $x<4$ (e.g. $x=0$): $\frac{3(0)-20}{0-4} = \frac{-20}{-4} = 5 \ge 0$ ✓. For $4<x<\frac{20}{3}$ (e.g. $x=5$): $\frac{15-20}{5-4} = \frac{-5}{1} = -5 \ge 0$? No ✗. For $x>\frac{20}{3}$ (e.g. $x=7$): $\frac{21-20}{7-4} = \frac{1}{3} \ge 0$ ✓ (and at $x=\frac{ …

Never cross-multiply an inequality by an expression (like x-4) whose sign is unknown; instead bring everything to one side, combine into a single fraction, and sign-test the resulting rational expression's numerator and denomina …

A very common error is to multiply both sides by (x−4) directly to 'clear the fraction' — since x−4 can be positive or negative depending on x, this can silently flip the inequality in only part of the domain, giving a wrong ans …