Which of the following relations is NOT always true?

Options

A. `abs(p+q) le absp + absq`
B. `abs(p-q) le absp - absq`
C. `absp - absq gt 0`, if `p^2 - q^2 gt 0`
D. `abs(p-q)^2 = (p-q)^2`

Answer

To determine which of the given relations is NOT always true, we will analyze each relation step by step.

### Step 1: Analyze Relation A
**Relation A:** |P + Q| ≤ |P| + |Q|

- This relation is known as the triangle inequality. It states that the absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.
- **Case 1:** If both P and Q are positive, then |P + Q| = P + Q and |P| + |Q| = P + Q. Thus, |P + Q| = |P| + |Q|.
- **Case 2:** If one of them is negative, say Q, then |P + Q| will still be less than or equal to |P| + |Q| because the absolute values will account for the negative.
- Therefore, this relation is always true.

### Step 2: Analyze Relation B
**Relation B:** |P - Q| ≤ |P| - |Q|

- Let's analyze this relation.
- **Case 1:** If P is greater than Q, then |P - Q| = P - Q and |P| - |Q| = P - Q (if both are positive). This holds true.
- **Case 2:** If Q is greater than P, then |P - Q| = Q - P and |P| - |Q| = |P| - Q. Here, |P| - Q could be negative if Q is significantly larger than P.
- Thus, |P - Q| can exceed |P| - |Q|, making this relation NOT always true.

### Step 3: Analyze Relation C
**Relation C:** |P| - |Q| > 0 if P² - Q² > 0

- This relation states that if P² > Q², then |P| > |Q|.
- If P² > Q², it implies that P > Q or P < -Q. In both cases, |P| will be greater than |Q|, confirming that this relation is always true.

### Step 4: Analyze Relation D
**Relation D:** |P - Q|² = (P - Q)²

- This relation states that the square of the absolute difference is equal to the square of the difference.
- This is true for all real numbers because squaring a number (positive or negative) results in a positive value.
- Therefore, this relation is always true.

### Conclusion
After analyzing all the relations, we find that Relation B is NOT always true.

### Final Answer
The relation that is NOT always true is **B: |P - Q| ≤ |P| - |Q|**.

---