Question

Which of the following is NOT true?

• A. \(|a + b| = |b + a|\)
• B. \(|a - b| = |b - a|\)
• C. \(|a + b| < |a| + |b|\)
• D. \(|a - b| > |a| - |b|\)

Hint:

Try a = b = 1 (or any two same-sign numbers) in each option; the strict less-than sign in |a+b| < |a| + |b| breaks down exactly when a and b share the same sign.

Answer 1

The Correct Option is C

Step 1: Recall what absolute value means.
\(|x|\) is the distance of x from zero on the number line, so it is always non-negative, and \(|x| = |-x|\) for any real x.

Step 2: Check option (1).
\(|a+b| = |b+a|\) is true because addition is commutative, \(a+b\) and \(b+a\) are the exact same number, so their absolute values must be equal. This statement is always true.

Step 3: Check option (2).
\(|a-b| = |b-a|\) is true because \(b - a = -(a-b)\), and the absolute value of a number equals the absolute value of its negative, \(|-(a-b)| = |a-b|\). This statement is always true.

Step 4: Check option (3).
The general rule, called the triangle inequality, is \(|a+b| \leq |a| + |b|\), that is, less than or equal to, not strictly less than.
Take a simple case: \(a = 1\), \(b = 1\). Then \(|a+b| = |2| = 2\), and \(|a| + |b| = 1 + 1 = 2\).
Here \(|a+b|\) equals \(|a|+|b|\), it is not strictly less. So the strict inequality in option (3) breaks down whenever a and b have the same sign (both positive or both negative).
This means option (3), as a general claim, is NOT always true.

Step 5: Check option (4).
The reverse triangle inequality says \(|a-b| \geq |a| - |b|\), so \(|a-b|\) is always at least as large as \(|a| - |b|\).
Testing values, say \(a=5, b=2\): \(|a-b| = 3\) and \(|a|-|b| = 3\); for \(a=2, b=5\): \(|a-b|=3\), \(|a|-|b|=-3\), so \(|a-b|\) stays comfortably ahead. This relation holds up as a true statement here.

Step 6: Conclusion.
Options (1), (2), and (4) hold as general true statements. Option (3) breaks down whenever a and b share the same sign, since then \(|a+b|\) equals \(|a|+|b|\) instead of being strictly smaller.
So option (3) is the one that is NOT true. \[ \boxed{(3)} \]