How many of the following relations is/are always true for any real values of p and q?
`abs(p+q) = absp + absq `
` abs(p-q) = absp - absq `
`abs((p)/q) = abs((p)/q)`

Options

A. 1. None
B. 2. one
C. 3. two
D. 4. three

Answer

To determine how many of the given relations are always true for any real values of \( p \) and \( q \), we will analyze each relation step by step.

### Step 1: Analyze the first relation
The first relation is:
\[ \text{abs}(p + q) = \text{abs}(p) + \text{abs}(q) \]
This relation is only true when both \( p \) and \( q \) are non-negative or both are non-positive. For example:
- Let \( p = 3 \) and \( q = 2 \):
\[ \text{abs}(3 + 2) = \text{abs}(5) = 5 \]
\[ \text{abs}(3) + \text{abs}(2) = 3 + 2 = 5 \]
This holds true.

- Now let \( p = 3 \) and \( q = -2 \):
\[ \text{abs}(3 - 2) = \text{abs}(1) = 1 \]
\[ \text{abs}(3) + \text{abs}(-2) = 3 + 2 = 5 \]
This does not hold true.

Thus, the first relation is **not always true**.

### Step 2: Analyze the second relation
The second relation is:
\[ \text{abs}(p - q) = \text{abs}(p) - \text{abs}(q) \]
This relation is also not always true. For example:
- Let \( p = 3 \) and \( q = -2 \):
\[ \text{abs}(3 - (-2)) = \text{abs}(3 + 2) = \text{abs}(5) = 5 \]
\[ \text{abs}(3) - \text{abs}(-2) = 3 - 2 = 1 \]
This does not hold true.

Thus, the second relation is **not always true**.

### Step 3: Analyze the third relation
The third relation is:
\[ \text{abs}\left(\frac{p}{q}\right) = \text{abs}\left(\frac{p}{q}\right) \]
This is an identity and is always true as it states that the absolute value of a number is equal to itself.

Thus, the third relation is **always true**.

### Conclusion
After analyzing all three relations, we find that only the third relation is always true for any real values of \( p \) and \( q \).

### Final Answer
Therefore, the number of relations that are always true is **1**.

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