Question:
Solve the given inequality for real x: $x+\frac{x}{2}+\frac{x}{3}<11$
Solution:
$x+\frac{x}{2}+\frac{x}{3}<11$
$\Rightarrow x\left(1+\frac{1}{2}+\frac{1}{3}\right)<11$
$\Rightarrow x\left(\frac{6+3+2}{6}\right)<11$
$\Rightarrow \frac{11 x}{6}<11$
$\Rightarrow \frac{11 x}{6 \times 11}<\frac{11}{11}$
$\Rightarrow \frac{x}{6}<1$
$\Rightarrow x<6$
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 $(-\infty, 6)$.