Решить неравенства.... Created by Giorgio. algebra-ru

Решить неравенства...

Giorgio @Giorgio

August 2021 1 102 Report

Решить неравенства...

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oganesbagoyan

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(Loq_2 (- Loq_2 x))² +Loq_2 (Loq_2 x)²  ≤ 3 .
ОДЗ : { x >0 ; - Loq_2 x > 0 ⇔{ x >0 ; Loq_2 x < 0 . 0<x <1.

(Loq_2 (- Loq_2 x))² +2Loq_2 |Loq_2 x|  ≤ 3 .
Loq_2 x <0
(Loq_2 (- Loq_2 x))² +  2Loq_2 (- Loq_2 x)   - 3  ≤ 0; *** t² +2t - 3 ≤ 0 ***
- 3 ≤ Loq_2 (- Loq_2 x)  ≤ 1 ;
осн. лог. =2 > 1 , поэтому
2^(-3) ≤ - Loq_2 x ≤ 2 ;
- 2 ≤  Loq_2 x ≤ -1/8;
1/4    ≤x 2^(-1/8) = стоит проверить арифметику
****************************************************************
4 ^(9x²/4) - (3x/2+1) ^((9x² -4)/4*1/Loq_2(3x/2 +1)) ≤ 3
4*4 ^(9x²-4)/4 -(3x/2+1) ^((9x² -4)/4*1/Loq_2(3x/2 +1)) ≤ 3
---------------------------------------------------------------------
обозначим  (3x/2+1) ^((9x² -4)/4*1/Loq_2(3x/2 +1)) = t ⇒
Loq_2 ( (3x/2+1) ^((9x² -4)/4*1/Loq_2(3x/2 +1)) =Loq_2 t;
(9x² -4)/4* 1/(Loq_2 (3x/2 +1)* Loq_2 (3x/2 +1) = Loq_2 t ;
(9x² -4)/4 = Loq _2 t ;
4 ^ (9x² -4)/4 =4 ^(loq_2 t) ;
4 ^ (9x² -4)/4 = 2 ^(2Loq_2 t);
4 ^ (9x² -4)/4 = 2 ^(Loq_2 t²);
4 ^ (9x² -4)/4 = t² ;
4t² +t -3 ≤ 0 ;
4(t +1)(t -3/4) ≤ 0;
-1≤ t ≤ 3/4

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