Question

If \(a\), \(b\) and \(c\) 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:

Test option (C) with small numbers like a = 1 and b = 2 to see it break, then confirm the other three follow directly from the triangle inequality.

Answer 1

The Correct Option is C

Step 1: Recall the basic triangle inequality.
For any two real numbers \(p\) and \(q\), the triangle inequality says:
\[ |p + q| \leq |p| + |q| \]
This is the one rule we lean on to check every option.

Step 2: Check option (A).
\(|a+b| \leq |a|+|b|\) is exactly the triangle inequality with \(p=a\) and \(q=b\). It is always true, for every real \(a\) and \(b\).

Step 3: Check option (B).
Write \(a - b\) as \(a + (-b)\). By the triangle inequality:
\[ |a-b| = |a + (-b)| \leq |a| + |-b| = |a| + |b| \]
So option (B) is also always true.

Step 4: Check option (C).
Try a simple pair of numbers to test this: let \(a = 1\) and \(b = 2\).
\[ |a - b| = |1 - 2| = 1 \]
\[ |a| - |b| = 1 - 2 = -1 \]
The claim says \(1 \leq -1\), which is false. So option (C) is not always true, in fact it fails completely here.
The correct general rule is the reverse triangle inequality, \(|a-b| \geq \big||a|-|b|\big|\), not the \(\leq\) version written in option (C).

Step 5: Check option (D).
Write \(a - c = (a-b) + (b-c)\), then apply the triangle inequality with \(p = a-b\) and \(q=b-c\):
\[ |a-c| = |(a-b)+(b-c)| \leq |a-b| + |b-c| \]
So option (D) is always true as well.

Final Answer:
Options (A), (B) and (D) hold for all real numbers, but option (C) fails, as shown by \(a=1, b=2\). So option (C) is the one that is NOT true. \[ \boxed{\text{Option (C)}} \]