x is an integer.
`{:("Quantity A","Quantity B"),((1)/(100^(x)),(1)/(99^(x))):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to compare the two quantities given:

**Quantity A:** \( \frac{1}{100^x} \)
**Quantity B:** \( \frac{1}{99^x} \)

We will analyze the relationship between these two quantities based on the value of \( x \), which is an integer.

### Step 1: Analyze when \( x > 0 \)

1. If \( x \) is a positive integer, we can say:
\[ 100^x > 99^x \]
This is because \( 100 > 99 \) and raising both sides to a positive power preserves the inequality.

2. Taking the reciprocal of both sides (remembering that taking the reciprocal reverses the inequality):
\[ \frac{1}{100^x} < \frac{1}{99^x} \]
Therefore, in this case:
\[ A < B \]

### Step 2: Analyze when \( x < 0 \)

1. If \( x \) is a negative integer, we can rewrite the quantities:
\[ A = \frac{1}{100^x} = 100^{-x} \quad \text{and} \quad B = \frac{1}{99^x} = 99^{-x} \]
Here, since \( -x \) is a positive integer, we can say:
\[ 100^{-x} < 99^{-x} \]
This is because \( 100 > 99 \) and raising both sides to a positive power preserves the inequality.

2. Therefore, taking the reciprocal again (which reverses the inequality):
\[ \frac{1}{100^x} > \frac{1}{99^x} \]
Thus, in this case:
\[ A > B \]

### Step 3: Analyze when \( x = 0 \)

1. If \( x = 0 \):
\[ A = \frac{1}{100^0} = 1 \quad \text{and} \quad B = \frac{1}{99^0} = 1 \]
Therefore:
\[ A = B \]

### Conclusion

From our analysis:
- When \( x > 0 \), \( A < B \).
- When \( x < 0 \), \( A > B \).
- When \( x = 0 \), \( A = B \).

Since the relationship between \( A \) and \( B \) changes based on the value of \( x \), we can conclude that the relationship cannot be determined.

### Final Answer
The relationship between Quantity A and Quantity B cannot be determined.

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