Solve the equation:
(i) `x+(x)/(2)+(x)/(3)lt11`
(ii) `(x)/(3)gt(x)/(2)+1`.

Answer

Let's solve the given inequalities step by step.

### Part (i): Solve the inequality \( x + \frac{x}{2} + \frac{x}{3} < 11 \)

1. **Combine the terms on the left side**:
To combine \( x \), \( \frac{x}{2} \), and \( \frac{x}{3} \), we first find a common denominator. The least common multiple (LCM) of 1, 2, and 3 is 6.

\[ x = \frac{6x}{6}, \quad \frac{x}{2} = \frac{3x}{6}, \quad \frac{x}{3} = \frac{2x}{6} \]

So, we rewrite the inequality:

\[ \frac{6x + 3x + 2x}{6} < 11 \]

This simplifies to:

\[ \frac{11x}{6} < 11 \]

2. **Cross-multiply to eliminate the fraction**:
Multiply both sides by 6 (since 6 is positive, the inequality sign remains the same):

\[ 11x < 66 \]

3. **Solve for \( x \)**:
Divide both sides by 11:

\[ x < 6 \]

Thus, the solution for the first part is:

\[ x \in (-\infty, 6) \]