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

This question tests how well we know the triangle inequality and its variations. The triangle inequality for real numbers p and q says $|p+q| \leq |p| + |q|$, and every option here is some version of that idea dressed up differently. Let's go through the four options one by one.

1. $|a+b| \leq |a|+|b|$: this is the triangle inequality stated directly, with no disguise. It holds for all real a and b, so this option is true.
2. $|a-b| \leq |a|+|b|$: replace b with -b in the triangle inequality. Since $|-b| = |b|$, we get $|a+(-b)| \leq |a| + |-b|$, which is exactly $|a-b| \leq |a|+|b|$. This holds for all real a and b.
3. $|a-b| \leq |a|-|b|$: check this with a=1, b=2. The left side is $|1-2|=1$. The right side is $|1|-|2| = 1-2 = -1$. The claim $1 \leq -1$ is false, so this option breaks down. The correct fact is the reverse triangle inequality, $|a-b| \geq ||a|-|b||$, which points the opposite way, so option (C) as written is simply not a true statement.
4. $|a-c| \leq |a-b|+|b-c|$: think of a, b, c as three points on the number line. The distance from a to c can never be more than going from a to b and then b to c. Writing $a-c = (a-b)+(b-c)$ and applying the triangle inequality confirms this is always true.

So three of the four options, (A), (B) and (D), are genuine inequalities that hold for every choice of real numbers, while (C) fails, as the counterexample a=1, b=2 shows directly.

Let's summarize:

• The triangle inequality $|p+q| \leq |p|+|q|$ is the base rule behind options (A), (B) and (D).
• The correct relation between $|a-b|$ and $|a|, |b|$ is $|a-b| \geq ||a|-|b||$, the reverse of what option (C) states.

So the statement that is NOT true is option (C).