Question

Решите неравенство 7-х-50х^2( в числителе)/х-7(в знаменателе ) -х^3-8х^2>=-1

Answer 1

$\frac{7-x-50x^2}{x-7}-x^3-8x^2\geq-1\\\\\frac{7-x-50x^2}{x-7}\geq x^3+8x^2-1\\\\7-x-50x^2\geq(x^3+8x^3-1)(x-7)\\7-x-50x^2\geq x^4+x^3-56x^2-x+7\\x^4+x^3-56x^2-x+7-7+x+50x^2\leq0\\x^4+x^3-6x^2\leq0\\x^2(x^2+x-6)\leq0\\x_1=x_2=0\\x^2+x-6=0\\D=1-4\cdot1\cdot(-6)=1+24=25\\x_{3,4}=\frac{-1\pm5}2\\x_3=-3,\;x_4=2\\\\x\in(-\infty;\;-3):\;x^4+x^3-6x^2>0\\x\in[-3;\;0]:\;x^4+x^3-6x^2<0\\x\in(0;\;2]:\;x^4+x^3-6x^2<0\\x\in(2;\;+\infty):\;x^4+x^3-6x^2>0\\\\OTBET:\;x\in[-3;\;2]$