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ashen
@ashen
August 2022
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Решите неравенство с логарифмами:
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yugolovin
Verified answer
Обозначим log_3 x=t;
(-8t-9)/(1-4t)≥t;
(8t+9)/(4t-1)-t≥0
;
(8t+9-4t^2+t)/(4t-1)≥0;
(4t^2-9t-9)/(4t-1)≤0;
(t-3)(4t+3)/(4t-1)≤0;
t∈(-∞;-3/4]∪(1/4;3].
1 случай. log_3 x≤ - 3/4; x∈(0; 3^(-3/4)]
2 случай 1/4< log_3 x≤3; x∈(3^(1/4);27]
Ответ: (0; 3^(-3/4)]∪
(3^(1/4);27]
2 votes
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Answers & Comments
Verified answer
Обозначим log_3 x=t;(-8t-9)/(1-4t)≥t;
(8t+9)/(4t-1)-t≥0;
(8t+9-4t^2+t)/(4t-1)≥0;
(4t^2-9t-9)/(4t-1)≤0;
(t-3)(4t+3)/(4t-1)≤0;
t∈(-∞;-3/4]∪(1/4;3].
1 случай. log_3 x≤ - 3/4; x∈(0; 3^(-3/4)]
2 случай 1/4< log_3 x≤3; x∈(3^(1/4);27]
Ответ: (0; 3^(-3/4)]∪ (3^(1/4);27]