Solve : `x + (x)/(2) + (x)/(3) gt 11`

Explanation

To solve the inequality \( x + \frac{x}{2} + \frac{x}{3} > 11 \), we will follow these steps:

### Step 1: Find a common denominator
The denominators in the expression are 1, 2, and 3. The least common multiple (LCM) of 2 and 3 is 6. We will rewrite each term with a denominator of 6.

### Step 2: Rewrite the inequality
We can express each term with the LCM:
\[ x = \frac{6x}{6}, \quad \frac{x}{2} = \frac{3x}{6}, \quad \frac{x}{3} = \frac{2x}{6} \]
Now substituting these into the inequality:
\[ \frac{6x}{6} + \frac{3x}{6} + \frac{2x}{6} > 11 \]

### Step 3: Combine the fractions
Now, we can combine the fractions on the left side:
\[ \frac{6x + 3x + 2x}{6} > 11 \]
This simplifies to:
\[ \frac{11x}{6} > 11 \]

### Step 4: Eliminate the denominator
To eliminate the fraction, multiply both sides of the inequality by 6 (note that 6 is positive, so the inequality direction remains the same):
\[ 11x > 66 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 11:
\[ x > 6 \]

### Step 6: Write the solution in interval notation
The solution can be expressed in interval notation as:
\[ x \in (6, \infty) \]

### Final Answer
Thus, the solution to the inequality \( x + \frac{x}{2} + \frac{x}{3} > 11 \) is:
\[ x \in (6, \infty) \]