Question

Solve the given inequality for real x: $x+\frac x2+\frac x3<11$.

Answer 1

$x+\frac x2+\frac x3<11$

⇒ $x(1+\frac 12+\frac 13) < 11$

⇒ $x(\frac {6+3+2}{6}) < 11$

⇒ $\frac {11x}{6} < 11$

⇒ $\frac {11x}{6\times 11} < \frac {11}{11}$

⇒ $\frac {x}{6} < 1$

⇒ $x< 6$
Thus, all real numbers x, which are less than 6, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, 6).