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Chapter 2: Problem 2

If \(a, b \in \mathbb{R}\), show that \(|a+b|=|a|+|b|\) if and only if \(a b \geq 0\).

Short Answer

Expert verified
The statement \(|a+b|=|a|+|b|\) if and only if \(ab \geq 0\) holds true and is proven.

Step by step solution

01

Direct Proof

Assume \(ab \geq 0\). This implies that either both \(a\) and \(b\) are nonnegative or both are nonpositive. In either case, their absolute values are the same as their original values (for nonnegative numbers) or the reverse of their original values (for nonpositive numbers). Thus, the equation \(|a+b|=|a|+|b|\) is equivalent to either \(a + b = a + b\) (for nonnegative numbers), which is trivially true, or \(-a - b = -a -b\) (for nonnegative numbers), which is also trivially true.
02

Contrapositive Proof

Assume \(|a+b|=|a|+|b|\). Note that the only way this equation can be true is if the sign of \(a + b\) is the same as the signs of \(a\) and \(b\). If \(a + b\) has the same sign as \(a\) (or equivalently, as \(b\)), then the product \(ab\) must be nonnegative, as multiplying numbers with the same sign results in a nonnegative product. Thus, the assumption that \(|a+b|=|a|+|b|\) implies that \(ab \geq 0\), as required.

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Most popular questions from this chapter

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