Question

If \( a, b, c \) are three real numbers, then which of the following is not true?

• A. \( |a + b| \leq |a| + |b| \)
• B. \( |a - b| \leq |a| + |b| \)
• C. \( |a - b| \leq |a| - |b| \)
• D. \( |a - c| \leq |a - b| + |b - c| \)

Hint:

Triangle inequalities always hold: \( |x + y| \leq |x| + |y| \), but subtracting absolute values isn't always valid.

Answer 1

The Correct Option is C

Option A: This is the triangle inequality: \( |a + b| \leq |a| + |b| \) — Always true.
Option B: Equivalent to triangle inequality again: \( |a - b| \leq |a| + |b| \) — Always true.
Option C: \( |a - b| \leq |a| - |b| \) is not always true. Consider \( a = 3, b = 5 \):
\[ |a - b| = |3 - 5| = 2,\quad |a| - |b| = 3 - 5 = -2 \Rightarrow 2 \leq -2 \text{ is False} \] Option D: This is the triangle inequality on three points: always true.
Hence, \[ {\text{(C) is not true}} \]