Решить неравенства а)lgx+6logₓ10≤5 b)lg²100x-5lgx больше 6. Created by marinka31. algebra-ru

1)lgx+6logx(10) <=5.................................ОДЗ:{x>0{x>1..................................lgx+6/ (lgx) <=5lgx+6/(lgx) - 5<=0[(lgx)^2 +6-5lgx] / lgx <=0Делаем замену:lgx=t, тогда:t^2-5t+6<=0t^2-5t+6=0D=(-5)^2-4*1*6=1t1=(5-1)/2=2; t2=(5+1)/2=3______+____[2]___-_____[3]___+______ //////////////////2<=t<=3Обратная замена:a)lgx>=2 lgx>=lg100 x>=100б) lgx<=3 lgx<=lg1000 x<=1000Ответ: x e [100;1000]2)(lg100x)^2 -5lgx>6 ОДЗ: x>0(lg100+lgx)^2 -5lgx-6>0(2+lgx)^2-5lgx-6>04+4lgx+ (lgx)^2-5lgx-6>0(lgx)^2-lgx-2>0Замена: lgx=t, тогда:t^2-t-2>0t^2-t-2=0D=(-1)^2-4*1*(-2)=9t1=(1-3)/2=-1; t2=(1+3)/2=2______+_____(-1)____-____(2)___+______///////////////////////// //////////////////////t<-1t>2Обратная замена:a)lgx<-1 lgx<lg1/10 x<1/10б) lgx>2 lgx>lg100 x>100С учетом ОДЗ: x e (0;1/10)U(100; + беск.) 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Решить неравенства а)lgx+6logₓ10≤5 b)lg²100x-5lgx больше 6...

marinka31 @marinka31

March 2022 1 31 Report

Решить неравенства а)lgx+6logₓ10≤5
b)lg²100x-5lgx больше 6

Answers & Comments

Gviona 1)lgx+6logx(10) <=5
.................................
ОДЗ:
{x>0
{x>1
..................................
lgx+6/ (lgx) <=5
lgx+6/(lgx) - 5<=0
[(lgx)^2 +6-5lgx] / lgx <=0
Делаем замену:
lgx=t, тогда:
t^2-5t+6<=0
t^2-5t+6=0
D=(-5)^2-4*1*6=1
t1=(5-1)/2=2; t2=(5+1)/2=3

______+____[2]___-_____[3]___+______
   //////////////////

2<=t<=3
Обратная замена:
a)lgx>=2
   lgx>=lg100
   x>=100
б) lgx<=3
    lgx<=lg1000
    x<=1000
Ответ: x e [100;1000]

2)(lg100x)^2 -5lgx>6 ОДЗ: x>0
(lg100+lgx)^2 -5lgx-6>0
(2+lgx)^2-5lgx-6>0
4+4lgx+ (lgx)^2-5lgx-6>0
(lgx)^2-lgx-2>0
Замена: lgx=t, тогда:
t^2-t-2>0
t^2-t-2=0
D=(-1)^2-4*1*(-2)=9
t1=(1-3)/2=-1; t2=(1+3)/2=2
______+_____(-1)____-____(2)___+______
///////////////////////// //////////////////////
t<-1
t>2
Обратная замена:
a)lgx<-1
   lgx<lg1/10
   x<1/10
б) lgx>2
   lgx>lg100
   x>100
С учетом ОДЗ: x e (0;1/10)U(100; + беск.)