Let Z denoted the set of integers, then
`{x inZ:|x-3|lt4}nn{x inZ:|x-4|lt5}=`

Options

A. `{-1,0,1,2,3,4}`
B. `{-1,0,1,2,3,4,5}`
C. `{,0,1,2,3,4,5,6}`
D. `{-1,0,1,2,3,4,5,6,7,8,9}`

Answer

To solve the problem, we need to find the intersection of two sets defined by the conditions involving absolute values. Let's denote the two sets as follows:

1. Set A: \( A = \{ x \in \mathbb{Z} : |x - 3| < 4 \} \)
2. Set B: \( B = \{ x \in \mathbb{Z} : |x - 4| < 5 \} \)

### Step 1: Solve for Set A
We start with the inequality for Set A:
\[ |x - 3| < 4 \]

This can be rewritten as two inequalities:
\[ -4 < x - 3 < 4 \]

Adding 3 to all parts of the inequality gives:
\[ -4 + 3 < x < 4 + 3 \]
\[ -1 < x < 7 \]

Since \( x \) must be an integer, we can list the integers in this range:
\[ x = 0, 1, 2, 3, 4, 5, 6 \]

Thus, Set A is:
\[ A = \{ 0, 1, 2, 3, 4, 5, 6 \} \]

### Step 2: Solve for Set B
Next, we solve for Set B:
\[ |x - 4| < 5 \]

This can also be rewritten as two inequalities:
\[ -5 < x - 4 < 5 \]

Adding 4 to all parts of the inequality gives:
\[ -5 + 4 < x < 5 + 4 \]
\[ -1 < x < 9 \]

Again, since \( x \) must be an integer, we can list the integers in this range:
\[ x = 0, 1, 2, 3, 4, 5, 6, 7, 8 \]

Thus, Set B is:
\[ B = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8 \} \]

### Step 3: Find the Intersection of Sets A and B
Now we find the intersection \( A \cap B \):
\[ A \cap B = \{ 0, 1, 2, 3, 4, 5, 6 \} \cap \{ 0, 1, 2, 3, 4, 5, 6, 7, 8 \} \]

The common elements in both sets are:
\[ A \cap B = \{ 0, 1, 2, 3, 4, 5, 6 \} \]

### Final Answer
Thus, the intersection of the two sets is:
\[ A \cap B = \{ 0, 1, 2, 3, 4, 5, 6 \} \]