"use strict"; var a = A.r.shuffle(['a', 'b', 'c', 'd', 'p', 'q', 'r', 'x', 'y', 'z']), b = a[0], c = a[1], d = A.r.sample([1, 2, 3, 4]), e = [[{ statement: "$".concat(b, "^2 < ").concat(c, "^{3}$"), explanation: "Statement $\\text{I}$ is always true. When $".concat(b, " < ").concat(d, "$ and $").concat(b, " < ").concat(c, "$, $").concat(b, "^2$ will be at most $").concat(d * d, "$ (and could be very small). Since $").concat(c, " > ").concat(b, "$, $").concat(c, "^{3}$ will always be greater than $").concat(b, "^2$ regardless of whether $").concat(b, "$ or $").concat(c, "$ are negative or positive.") }, { statement: "$".concat(b, "^2 < ").concat(c, "^{4}$"), explanation: "Statement $\\text{I}$ is always true. When $".concat(b, " < ").concat(d, "$ and $").concat(b, " < ").concat(c, "$, $").concat(b, "^2$ will be at most $").concat(d * d, "$ (and could be very small). Since $").concat(c, " > ").concat(b, "$, $").concat(c, "^{4}$ will always be greater than $").concat(b, "^2$ regardless of whether $").concat(b, "$ or $").concat(c, "$ are negative or positive.") }], [{ statement: "$".concat(b, "^{2} < ").concat(c, "^{3}$"), explanation: "Statement $\\text{II}$ is always true. When $".concat(b, " < ").concat(d, "$ and $").concat(b, " < ").concat(c, "$, raising $").concat(b, "$ to power 2 makes it smaller if positive, while $").concat(c, "^{3}$ remains larger. For negative values, since 3 > 2, the inequality still holds because the higher exponent on $").concat(c, "$ ensures $").concat(c, "^{3} > ").concat(b, "^{2}$.") }, { statement: "$".concat(b, "^{3} < ").concat(c, "^{5}$"), explanation: "Statement $\\text{II}$ is always true. When $".concat(b, " < ").concat(d, "$ and $").concat(b, " < ").concat(c, "$, raising $").concat(b, "$ to power 3 makes it smaller if positive, while $").concat(c, "^{5}$ remains larger. For negative values, since $5 > 3$, the inequality still holds because the higher exponent on $").concat(c, "$ ensures $").concat(c, "^{5} > ").concat(b, "^{3}$.") }], [{ statement: "$".concat(b, "^{3} < ").concat(c, "^{2}$"), explanation: "Statement $\\text{III}$ is always true. When $".concat(b, " < 0$, $").concat(b, "^{3}$ is negative (since $3$ is odd), while $").concat(c, "^{2}$ is always non-negative (since 2 is even). When $0 \\leq ").concat(b, " < ").concat(d, "$, we have $").concat(b, "^{3} < ").concat(b, " < ").concat(c, " < ").concat(c, "^{2}$.") }, { statement: "$".concat(b, "^{5} < ").concat(c, "^{4}$"), explanation: "Statement $\\text{III}$ is always true. When $".concat(b, " < 0$, $").concat(b, "^{5}$ is negative (since $5$ is odd), while $").concat(c, "^{4}$ is always non-negative (since 4 is even). When $0 \\leq ").concat(b, " < ").concat(d, "$, we have $").concat(b, "^{5} < ").concat(b, " < ").concat(c, " < ").concat(c, "^{4}$.") }]], f = [[{ statement: "$".concat(b, "^2 < ").concat(c, "$"), explanation: "Statement $\\text{I}$ is not always true. For example, if $".concat(b, " = -10$ and $").concat(c, " = -5$, then $").concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$, but $").concat(b, "^2 = 100$ which is greater than $").concat(c, " = -5$.") }, { statement: "$".concat(b, "^3 < ").concat(c, "$"), explanation: "Statement $\\text{I}$ is not always true. For example, if $".concat(b, " = -4$ and $").concat(c, " = -2$, then $").concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$, but $").concat(b, "^3 = -64$ which is less than $").concat(c, " = -2$.") }], [{ statement: "$".concat(b, "^{2} < ").concat(c, "^{3}$"), explanation: "Statement $\\text{II}$ is not always true. For example, if $".concat(b, " = -3$ and $").concat(c, " = -2$, then $").concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$, but $").concat(b, "^{2} = 9$ while $").concat(c, "^{3} = -8$, so $").concat(b, "^{2} > ").concat(c, "^{3}$.") }, { statement: "$".concat(b, "^{4} < ").concat(c, "^{3}$"), explanation: "Statement $\\text{II}$ is not always true. For example, if $".concat(b, " = -2$ and $").concat(c, " = -1$, then $").concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$, but $").concat(b, "^{4} = 16$ while $").concat(c, "^{3} = -1$, so $").concat(b, "^{4} > ").concat(c, "^{3}$.") }], [{ statement: "$".concat(b, "^4 < ").concat(c, "^2$"), explanation: "Statement $\\text{III}$ is not always true. For example, if $".concat(b, " = -3$ and $").concat(c, " = -2$, then $").concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$, but $").concat(b, "^4 = 81$ while $").concat(c, "^2 = 4$, so $").concat(b, "^4 > ").concat(c, "^2$.") }, { statement: "$".concat(b, "^6 < ").concat(c, "^3$"), explanation: "Statement $\\text{III}$ is not always true. For example, if $".concat(b, " = -2$ and $").concat(c, " = -1$, then $").concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$, but $").concat(b, "^6 = 64$ while $").concat(c, "^3 = -1$, so $").concat(b, "^6 > ").concat(c, "^3$.") }]]; //? //? //? //? var g = [,,,].fill(''), h = [,,,].fill(''); var i = A.r.sample(e[0]), j = A.r.sample(e[1]), k = A.r.sample(e[2]), l = A.r.sample(f[0]), m = A.r.sample(f[1]), n = A.r.sample(f[2]), o = A.r.intInclusive(0, 2); //? //? //? //? //? //? g[0] = o === 0 ? i.statement : l.statement, h[0] = o === 0 ? i.explanation : l.explanation, g[1] = o === 1 ? j.statement : m.statement, h[1] = o === 1 ? j.explanation : m.explanation, g[2] = o === 2 ? k.statement : n.statement, h[2] = o === 2 ? k.explanation : n.explanation; var p = "Which of the following is always true for the numbers $".concat(b, "$ and $").concat(c, "$, if $").concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$?\n\n$\\text{I}$. ").concat(g[0], "\n\n$\\text{II}$. ").concat(g[1], "\n\n$\\text{III}$. ").concat(g[2]), q = o === 0 ? '$\\text{I}$ only' : o === 1 ? '$\\text{II}$ only' : '$\\text{III}$ only', r = q, s = ['$\\text{I}$ only', '$\\text{II}$ only', '$\\text{III}$ only', '$\\text{I}$ and $\\text{II}$ only', '$\\text{I}$ and $\\text{III}$ only', '$\\text{II}$ and $\\text{III}$ only', '$\\text{I}$, $\\text{II}$, and $\\text{III}$'], t = s.filter(function (b) { return b !== q; }), u = _.sampleSize(t, 3), v = "This is as much a number properties problem as it is an inequalities problem. We have two inequalities: $".concat(b, " < ").concat(c, "$ and $").concat(b, " < ").concat(d, "$. These tell us that $").concat(b, "$ could be negative, zero, or a positive fraction less than $").concat(d, "$, and $").concat(c, "$ is greater than $").concat(b, "$ but could still be negative.\n\nLet's analyze each statement:\n\n").concat(h[0], "\n\n").concat(h[1], "\n\n").concat(h[2], "\n\nSince only Statement $\\text{").concat(o === 0 ? 'I' : o === 1 ? 'II' : 'III', "}$ is always true, the answer is \"").concat(q, "\"."); //? //? //? //? ({ prompt: p, answer: r, distractors: u, explanation: v });

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