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  "body": "\n## Statement\n\nFor all real numbers $a$ and $b$,\n$$|a + b| \\le |a| + |b|,$$\nwhere $|\\cdot|$ is the [[absolute value|00S7C68D51]].\n\n## Remarks\n\nProof: $-|a| \\le a \\le |a|$ and $-|b| \\le b \\le |b|$. Add these inequalities to get $-(|a| + |b|) \\le a + b \\le |a| + |b|$.\nEquality holds exactly when $a b \\ge 0$.\nFor distances, it gives $|x - z| \\le |x - y| + |y - z|$. The [[reverse triangle inequality|0C9S13WKMF]] is a consequence.\n\n## Explanations\n\n### L1 human\n\nThe name comes from geometry: one side of a triangle is never longer than the other two sides together.\nFor numbers, $|a + b|$ can be smaller than $|a| + |b|$ when $a$ and $b$ have opposite signs, for example $|5 + (-3)| = 2 \u003c 8$.\n",
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