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marinka31
@marinka31
August 2022
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Решить уравнение 2(lg2-1)+lg([tex] 5^{ \sqrt{x} } [/tex]+1)≤lg([tex] 5^{1- \sqrt{x} } [/tex]+5)
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oganesbagoyan
Verified answer
2(Lq2 - 1)+Lq(5^x +1) ≤Lq
(
5^(1-√x) +5
)
;
2*(Lq2 -Lq10)+Lq(5^x +1) ≤Lq
(
5^(1-√x) +5
)
;
2*(Lq2/10)+Lq(5^x +1) ≤Lq
(
5^(1-√x) +5
)
;
Lq5^(-2)+Lq(5^x +1) ≤ Lq
(
5^(1-√x) +5
)
;
Lq (5^x +1)/25 ≤ Lq
(
5^(1-√x) +5
) ;
0 ≤ (5^x +1)/25 ≤ 5/(5^√x) +5 ;* * * t =5^x * * *
* * * (5^x +1)/25 ≥ 0 при всех x .
0 ≤ (t+1)/25 ≤ 5/t +5 ;
0 ≤ t² +t ≤ 125 +125t ;
t² -124t -125 ≤ 0 ;
(t +1)(t-125) ≤ 0 ;
-1 ≤ t ≤ 125 ;
-1 ≤ 5^x ≤ 5³;
x ∈(-∞ ;3].
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Answers & Comments
Verified answer
2(Lq2 - 1)+Lq(5^x +1) ≤Lq(5^(1-√x) +5 ) ;2*(Lq2 -Lq10)+Lq(5^x +1) ≤Lq(5^(1-√x) +5 ) ;
2*(Lq2/10)+Lq(5^x +1) ≤Lq(5^(1-√x) +5 ) ;
Lq5^(-2)+Lq(5^x +1) ≤ Lq(5^(1-√x) +5 ) ;
Lq (5^x +1)/25 ≤ Lq(5^(1-√x) +5 ) ;
0 ≤ (5^x +1)/25 ≤ 5/(5^√x) +5 ;* * * t =5^x * * *
* * * (5^x +1)/25 ≥ 0 при всех x .
0 ≤ (t+1)/25 ≤ 5/t +5 ;
0 ≤ t² +t ≤ 125 +125t ;
t² -124t -125 ≤ 0 ;
(t +1)(t-125) ≤ 0 ;
-1 ≤ t ≤ 125 ;
-1 ≤ 5^x ≤ 5³;
x ∈(-∞ ;3].