Solve : `17 lt 9y -8l y in Z`

Answer

To solve the inequality \( 17 < 9y - 81 \) where \( y \) belongs to the set of integers \( \mathbb{Z} \), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the original inequality:
\[ 17 < 9y - 81 \]

### Step 2: Isolate the Variable
To isolate \( y \), we first need to move the constant term (-81) to the other side of the inequality. We do this by adding 81 to both sides:
\[ 17 + 81 < 9y \]
This simplifies to:
\[ 98 < 9y \]

### Step 3: Divide by 9
Next, we divide both sides of the inequality by 9 to solve for \( y \):
\[ \frac{98}{9} < y \]
This can also be written as:
\[ y > \frac{98}{9} \]

### Step 4: Calculate \( \frac{98}{9} \)
Now we calculate \( \frac{98}{9} \):
\[ \frac{98}{9} \approx 10.888\ldots \]
This means that \( y \) must be greater than approximately 10.89.

### Step 5: Determine Integer Solutions
Since \( y \) must be an integer (as \( y \in \mathbb{Z} \)), the smallest integer greater than \( 10.888\ldots \) is 11. Therefore, the integer solutions for \( y \) are:
\[ y = 11, 12, 13, \ldots \]

### Step 6: Write the Solution Set
We can express the solution set in set notation:
\[ y \in \{11, 12, 13, \ldots\} \]

### Final Answer
Thus, the solution to the inequality \( 17 < 9y - 81 \) where \( y \) belongs to the integers is:
\[ y \in \{11, 12, 13, \ldots\} \]

---