`|x-7|lt9`

Answer

To solve the inequality \( |x - 7| < 9 \), we will break it down into two separate cases based on the definition of absolute value.

### Step 1: Understand the definition of absolute value
The expression \( |x - 7| < 9 \) means that the distance between \( x \) and \( 7 \) is less than \( 9 \). This can be expressed as:
\[ -9 < x - 7 < 9 \]

### Step 2: Split the inequality into two parts
We can split this compound inequality into two separate inequalities:
1. \( x - 7 < 9 \)
2. \( x - 7 > -9 \)

### Step 3: Solve the first inequality
Starting with the first inequality:
\[ x - 7 < 9 \]
Add \( 7 \) to both sides:
\[ x < 9 + 7 \]
This simplifies to:
\[ x < 16 \]

### Step 4: Solve the second inequality
Now, we solve the second inequality:
\[ x - 7 > -9 \]
Again, add \( 7 \) to both sides:
\[ x > -9 + 7 \]
This simplifies to:
\[ x > -2 \]

### Step 5: Combine the results
Now we combine the results from both inequalities:
\[ -2 < x < 16 \]

### Final Answer
Thus, the solution to the inequality \( |x - 7| < 9 \) is:
\[ (-2, 16) \]