Question

Решите неравенства: 3^(2x-5)+3^(2x-6)-3^(2x-7)-3^(2x-8)≤32 5^4x-5^(4x-1)+5^(4x-2)-5^(4x-3)≥104

Answer 1

$1)$

$3^{2x-5} + 3^{2x-6} - 3^{2x-7} - 3^{2x-8} leq 32$

$3^{2x-5}(1+ 3^{-1} - 3^{-2} - 3^{-3}) leq 32$

$3^{2x-5}(1+frac{1}{3} } - frac{1}{9} - frac{1}{27} ) leq 32$

$3^{2x-5}(1+ frac{9}{27} } - frac{3}{27} - frac{1}{27} ) leq 32$

$3^{2x-5}*1 frac{5}{27} } leq 32$

$3^{2x-5}*frac{32}{27} } leq 32$

$3^{2x-5} leq 32* frac{27}{32}$

$3^{2x-5} leq 27$

$3^{2x-5} leq 3^3$

$2x-5 leq 3$

$2x leq 8$

$x leq 4$

-------------[4]-----------
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Ответ: $(-$ ∞ $;4]$

$2)$

$5^{4x} - 5^{4x-1} + 5^{4x-2} - 5^{4x-3} geq 104$

$5^{4x} (1- 5^{-1} + 5^{-2} - 5^{-3}) geq 104$

$5^{4x} (1- frac{1}{5} + frac{1}{25} - frac{1}{125} ) geq 104$

$5^{4x} (1- frac{25}{125} + frac{5}{125} - frac{1}{125} ) geq 104$

$5^{4x} * frac{104}{125} geq 104$

$5^{4x} geq 104* frac{125}{104}$

$5^{4x} geq 125$

$5^{4x} geq 5^3$

$4x geq 3$

$x geq 0.75$

----------[0.75]-------------
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Ответ:  $[0.75;+$ ∞ $)$