{
  "problem": "Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \\neq 0$. $r = \\dfrac{t^2 + 3t - 28}{-3t^2 + 18t + 81} \\times \\dfrac{4t - 36}{t + 7} $",
  "hints": [
    "First factor out any common factors.",
    "$r = \\dfrac{t^2 + 3t - 28}{-3(t^2 - 6t - 27)} \\times \\dfrac{4(t - 9)}{t + 7} $",
    "Then factor the quadratic expressions.",
    "$r = \\dfrac {(t + 7)(t - 4)} {-3(t - 9)(t + 3)} \\times \\dfrac {4(t - 9)} {t + 7} $",
    "Then multiply the two numerators and multiply the two denominators.",
    "$r = \\dfrac { (t + 7)(t - 4) \\times 4(t - 9)} { -3(t - 9)(t + 3) \\times (t + 7)} $",
    "$r = \\dfrac {4(t + 7)(t - 4)(t - 9)} {-3(t - 9)(t + 3)(t + 7)} $",
    "Notice that",
    "$(t - 9)$",
    "and",
    "$(t + 7)$",
    "appear in both the numerator and denominator so we can cancel them.",
    "$r = \\dfrac {4(t + 7)(t - 4)\\cancel{(t - 9)}} {-3\\cancel{(t - 9)}(t + 3)(t + 7)} $",
    "We are dividing by",
    "$t - 9$",
    ", so",
    "$t - 9 \\neq 0$",
    "Therefore,",
    "$t \\neq 9$",
    "$r = \\dfrac {4\\cancel{(t + 7)}(t - 4)\\cancel{(t - 9)}} {-3\\cancel{(t - 9)}(t + 3)\\cancel{(t + 7)}} $",
    "We are dividing by",
    "$t + 7$",
    ", so",
    "$t + 7 \\neq 0$",
    "Therefore,",
    "$t \\neq -7$",
    "$r = \\dfrac {4(t - 4)} {-3(t + 3)} $",
    "$ r = \\dfrac{-4(t - 4)}{3(t + 3)}; t \\neq 9; t \\neq -7 $"
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/274/1608014049.json'}