Решить неравенства:1)2^x = 1/82)(1/2)^(3x-5) больше или равно 43)log2 x > 24)log0,2(x+2)...

Question

Решить неравенства:
1)2^x = 1/8
2)(1/2)^(3x-5) больше или равно 4
3)log2 x > 2
4)log0,2(x+2) > -1
5)(1/9)^x - 6 * (1/3)^x > - 9
6)(log0,5)^2 + log0,5 x - 3 > 0
7)log3 4,5
_________ > 1
3 - log3 x
8)9^x - 2 * 3^x + 1
________ > 0
9^x - 2 * 3^x + 2
9)(2 - корень из 3)^2 - 4 * ( 1 ) +1 < 0
_____________
2 +корень из 3
10)4^(x+2) - 13 * 4^x > 12

Answer 1

1)

x= -3
2)

-3x+5 ≥ 2
-3x ≥ -3
x≤ 1
3) log₂x>2
ОДЗ: x>0
log₂x > log₂4
x > 4
Ответ: x>4
4) log₀.₂ (x+2) > -1
ОДЗ: x+2>0 ⇒ x>-2
log₀.₂(x+2) > log₀.₂ 5
x+2 < 5
x < 3
x∈ (-2;3)
Ответ: (-2;3)
5) 0 " alt=" ( \frac{1}{9}) ^{x}-6* ( \frac{1}{3}) ^{x}+9>0 " align="absmiddle" class="latex-formula">
0 " alt=" ( \frac{1}{3}) ^{x}=t; t>0 " align="absmiddle" class="latex-formula">
t² - 6t + 9>0
D₁ = 9-9 = 0
t = 3
++++++ (3) +++++
0} \atop {t \neq 3}} \right. " alt=" \left \{ {{t>0} \atop {t \neq 3}} \right. " align="absmiddle" class="latex-formula">
0 } \atop { ( \frac{1}{3}) ^{x} \neq 3 }} \right. " alt=" \left \{ {{ ( \frac{1}{3}) ^{x}>0 } \atop { ( \frac{1}{3}) ^{x} \neq 3 }} \right. " align="absmiddle" class="latex-formula">
x ≠ -1
Ответ: x≠ -1
7) 1" alt=" \frac{ log_{3}4.5 }{ 3- log_{3}x } > 1" align="absmiddle" class="latex-formula">
ОДЗ: x> 0
3- log_{3}x } \atop {3- log_{3}x >0}} \right. " alt=" \left \{ {{ log_{3}4.5>3- log_{3}x } \atop {3- log_{3}x >0}} \right. " align="absmiddle" class="latex-formula">

log₃4.5 > 3-log₃x
log₃4.5 + log₃x > 3
log₃4.5x > 3
4.5x > 27
x > 6

3-log₃x > 0
-log₃x > -3
log₃x < 3
x < 27
x ∈ (6; 27)
Ответ: (6;27)
10) 12 " alt=" 4^{x+2}-13* 4^{x}> 12 " align="absmiddle" class="latex-formula">
12 " alt="16* 4^{x}-13* 4^{x}> 12 " align="absmiddle" class="latex-formula">
3* 12 " alt=" 4^{x} > 12 " align="absmiddle" class="latex-formula">
4 " alt=" 4^{x}> 4 " align="absmiddle" class="latex-formula">
x>1
Ответ: x>1