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Задать вопрос экспертам

Консультация № 188454

19.12.2015, 00:09

0.00 руб.

0 1 1

Образование

Уважаемые эксперты! Пожалуйста, помогите решить неравенство:
((5х-2)^2)/(x-3)>=(4-20x+25x^2)/(24-11x+x^2)

Обсуждение

давно

Roman Chaplinsky / Химик CH

Модератор
156417
2175

19.12.2015, 01:04

общий

это ответ

Здравствуйте, Посетитель - 399097!
В первую очередь заметим, что числители равны
(5x-2)²=25x²-20x+4
они обращаются в 0 при x=2/5 и при этом неравенство выполняется 0[$8805$]0

Для всех x[$8800$]2/5 числители заведомо положительны и мы можем разделить обе стороны неравенства на (5x-2)²
1/(x-3)[$8805$]1/(x²-11x+24)
1/(x-3)[$8805$]1/((x-3)(x-8))
при x=3 оба знаменателя обращаются в 0 и выражения не имеют смысла

при x>3 умножаем обе части на положительное x-3
1[$8805$]1/(x-8)
знаменатель правой части обращается в 0 при x=8
при 3<x<8 правая часть 1/(x-8)<0[$8804$]1 и неравенство выполняется

при x>8 умножаем обе стороны на положительное x-8
x-8[$8805$]1
и неравенство выполняется при x[$8805$]9

при x<3 умножаем на отрицательное x-3
1[$8804$]1/(x-8)
при этом правая часть заведомо отрицательна 1/(x-8)<0<1 и неравенство не выполняется

Объединяем решения
x[$8712$]{2/5}[$8746$](3; 8)[$8746$][9; [$8734$])

Форма ответа

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