$\begin{array}{1 1} (A)\;(-3,\infty)\\(B)\;(-\infty,6)\\(C)\;(-4,\infty)\\(D)\; (-9,\infty)\end{array} $

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- Same Quantity can be added (a subtracted ) to (from ) both sides of the inequality with out changing the sign of the in equality.
- Same positive quantities can be multiplied or divided to both side of the in equality with out changing the sign of the inequality.
- If same negative quantity is multiplied or divided to both sides of the inequality is reversed i.e $ '>'$ sign changes to $'<' $ and $'<'$ changes $'>'$ .

The given inequality is $x + \large\frac{x}{2} +\frac{x}{3} $$< 11$

$=> x [1+ \large\frac{1}{2} +\frac{1}{3} \bigg] $$<11$

$=> x \bigg[ \large\frac{6+ 3+2}{6} \bigg]$$ <11$

$=> \large \frac{11 x}{6} $$<11$

Dividing both sides of the inequality by 11.

=> $ \large\frac{11 x}{6 \times 11} < \frac{11}{11}$

=> $ \large\frac{x}{6} $$<1$

=> $ x < 6$

Step 2:

All real numbers which are less than 6 satisfy the given in equality.

The solution set is $(-\infty, 6)$

Hence B is the correct answer.

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