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  "body": "\n## Definition\n\nLet $x$ be a real number.\nThe absolute value of $x$ is the real number\n$$|x| = \\begin{cases} x \u0026 \\text{if } x \\ge 0, \\\\ -x \u0026 \\text{if } x \u003c 0. \\end{cases}$$\n\n## Remarks\n\nFor each real $x$: $|x| \\ge 0$; $|x| = 0$ only for $x = 0$; $|-x| = |x|$; and $|x| = \\max(x, -x)$.\nThe distance between two real numbers $x$ and $y$ is $|x - y|$.\nFor $r \\ge 0$, the inequality $|x| \\le r$ holds exactly when $-r \\le x \\le r$.\n\n## Explanations\n\n### L1 human\n\nThe absolute value removes the minus sign from a number.\nOn the number line, $|x|$ is the distance from $x$ to $0$, and $|x - y|$ is the distance from $x$ to $y$.\n",
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