If `A={x: |x|lt2},B={x: |x-5|le2}`,
`C={x: |x|gtx}andD={x: |x|ltx}`
The number of integral values in `AuuB` is

Options

A. 4
B. 6
C. 8
D. 10

Explanation

To solve the problem, we need to find the number of integral values in the union of two sets \( A \) and \( B \).

**Step 1: Define the set \( A \)**
Set \( A \) is defined as:
\[ A = \{ x : |x| < 2 \} \]
This means \( x \) must lie between -2 and 2. Therefore, we can express this as:
\[ -2 < x < 2 \]
The integral values in this range are:
\[ -1, 0, 1 \]

**Step 2: Define the set \( B \)**
Set \( B \) is defined as:
\[ B = \{ x : |x - 5| \leq 2 \} \]
This can be rewritten as:
\[ -2 \leq x - 5 \leq 2 \]
Adding 5 to all parts of the inequality gives:
\[ 3 \leq x \leq 7 \]
The integral values in this range are:
\[ 3, 4, 5, 6, 7 \]

**Step 3: Find the union of sets \( A \) and \( B \)**
The union \( A \cup B \) combines all unique elements from both sets:
\[ A \cup B = \{-1, 0, 1\} \cup \{3, 4, 5, 6, 7\} \]
Thus, the union is:
\[ A \cup B = \{-1, 0, 1, 3, 4, 5, 6, 7\} \]

**Step 4: Count the integral values in \( A \cup B \)**
Now we count the integral values in the union:
\[ -1, 0, 1, 3, 4, 5, 6, 7 \]
There are a total of 8 integral values.

**Final Answer:**
The number of integral values in \( A \cup B \) is \( 8 \).

---