7) log₀,₅(x + 8) - log₀,₅(x - 3) > log₀,₅3x;
ОДЗ: x > -8; Имеем: x > 3.
x > 3;
x > 0
log₀,₅(x + 8) > log₀,₅3x + log₀,₅(x - 3);
log₀,₅(x + 8) > log₀,₅3x(x - 3);
x + 8 < 3x(x - 3);
3x² - 9x - x - 8 > 0;
3x² - 10x - 8 > 0;
3x² - 10x - 8 = 0; D = 100 + 96 = 196; √D = 14;
x₁ = (10 + 14)/6 = 4; x₂ = (10 - 14)/6 = -4/6 = -2/3
------ ++++
---------------------3----------------4----------->
x∈(4; ∞).
Ответ: (4; ∞).
8) log²₃(27x) + log₃(x³/9) = 17;
ОДЗ: x > 0
(log₃27 + log₃x)² + log₃(x³) - log₃9 = 17;
(3 + log₃x)² + 3log₃x - 2 - 17 = 0;
9 + 6log₃x + log²₃x + 3log₃x - 19 = 0;
log²₃x + 9log₃x - 10 = 0. Замена: log₃x = t
t² + 9t - 10 = 0;
t₁ = -10; t₂ = 1.
Обратная замена:
log₃x = -10 или log₃x = 1
x₁ = 3⁻¹⁰ x₂ = 3
Ответ: 3⁻¹⁰; 3.
2. x²-2x-8=0
D=4+4*8=36
x₁=<u>2-6</u>=-2
2
x₂=<u>2+6</u>=4
2
x₁*x₂=-2*4= -8
Ответ: -8
3. 5(х-6)(х+7)<0
(x-6)(x+7)<0
x=6 x=-7
+ - +
------- -7----------- 6----------
\\\\\\\\\\
x∈(-7; 6)
Ответ: (-7; 6)
4) {2x+15>0
{7x-14>0
{2x>-15
{7x>14
{x>-7.5
{x>2
\\\\\\\\\\\\\\\\\\\\\\\
----- -7.5 ------- 2 ----------
\\\\\\\\\\\\\\\\\\\\\
x>2 или х∈(2; +∞)