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

Options

A. 1
B. 2
C. 3
D. 0

Explanation

To solve the problem, we need to analyze the sets \( A \) and \( C \) and then find the number of integral values in the union of these sets.

### Step 1: Define Set A
Set \( A \) is defined as:
\[ A = \{ x : |x| < 2 \} \]
This means that \( x \) must be between -2 and 2, but not including -2 and 2. Therefore, we can express this as:
\[ A = (-2, 2) \]

### Step 2: Identify Integral Values in Set A
The integral values in the interval \((-2, 2)\) are:
- \(-1\)
- \(0\)
- \(1\)

Thus, the integral values in set \( A \) are:
\[ \{-1, 0, 1\} \]

### Step 3: Define Set C
Set \( C \) is defined as:
\[ C = \{ x : |x| > x \} \]
We need to analyze this condition. The expression \( |x| > x \) holds true for negative values of \( x \) because:
- If \( x \) is negative, \( |x| = -x \) and thus \( -x > x \) (which is true).
- If \( x \) is non-negative (i.e., \( x \geq 0 \)), then \( |x| = x \) and \( x > x \) is false.

Therefore, set \( C \) consists of all negative numbers:
\[ C = (-\infty, 0) \]

### Step 4: Identify Integral Values in Set C
The integral values in set \( C \) (which includes all negative integers) are:
\[ \{-1, -2, -3, \ldots\} \]
However, since we are interested in the union with set \( A \), we will focus on the integral values that are also in \( A \).

### Step 5: Find the Union of Sets A and C
Now we find the union of sets \( A \) and \( C \):
\[ A \cup C = \{-1, 0, 1\} \cup (-\infty, 0) \]
This union includes all negative integers and the values from set \( A \):
\[ A \cup C = \{-1, 0, 1\} \cup \{-1, -2, -3, \ldots\} \]
Thus, the integral values in \( A \cup C \) are:
\[ \{-1, -2, -3, \ldots, 0, 1\} \]

### Step 6: Count the Integral Values
The integral values in \( A \cup C \) are:
- All negative integers (which are infinite)
- \(0\)
- \(1\)

Since the negative integers are infinite, we conclude that the number of integral values in \( A \cup C \) is infinite.

### Final Answer
The number of integral values in \( A \cup C \) is infinite.

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