Question

Log5(2-x)+0,5log5(4x-11)^2=0
Lgx^2+lg(x+4)^2>=-lg1/9

Answer 1

log(5)(2-x)+0,5log(5)(4x-11)=0
{2-x>0⇒x<2
{4x-11≠0⇒x≠2,75
x∈(-∞;2)
log(5)(2-x)+log(5)√(4x-11)²=0
log(5)(2-x)+log(5)|4x-11|=0
log(5)[(2-x)*|4x-11|]=0
(2-x)*|4x-11|=1
x∈(-∞;2)⇒|4x-11|=11-4x
(2-x)(11-4x)=0
x=2не удов усл
х=2,75 не удов усл
Ответ нет решения

lgx²+lg(x+4)²≥-lg1/9
{x≠0
{x≠-4
x∈(-∞;-4) U (-4;0) U (0;∞)
lg[x²(x+4)²]≥lg9
x²(x+4)²≥9
x²(x+4)²-9≥0
(x(x+4)-3)(x(x+4)+3)≥0
(x²+4x-3)(x²+4x+3)≥0
x²+4x-3=0
D=16+12=28
x1=(-4-2√7)/2=-2-√7 U x2=-2+√7
x²+4x+3=0
x1+x2=-4 U x1*x2=3⇒x1=-3 U x2=-1
           +                  _                    +                      _                  +
-------------[-2-√7]------(-4)-------[-3]------------[-1]------(0)--------[-2+√7]--------------
////////////////////////////////////\\\\\\\\\\\\\\\\\\\\\\/////////////////////////////////
x∈(-∞;-2-√7] U [-3;-1] U [-2+√7;∞)

Answer 2

$1); ; log_5(2-x)+0,5cdot log_5(4x-11)^2=0; ;; ; ODZ:; left { {{x textless 2} atop {xne frac{11}{4}}} right. ; ,; x textless 2\\log_2(2-x)+log_5(4x-11)=0; ,; ; ; 0=log_51\\(2-x)(4x-11)=1\\8x-22-4x^2+11x=1\\4x^2-19x+23=0\\D=19^2-4cdot 4cdot 23=-7 textless 0; ; to ; ; net; reshenij$

$2); ; lgx^2+lg(x+4)^2 geq -lgfrac{1}{9}; ,; ; ODZ:; ; left { {{xne 0} atop {xne -4}} right. \\lg(x^2(x+4)^2) geq lg9 \\x^2(x+4)^2 geq 9\\Big (x(x+4)Big )^2-3^2 geq 0\\(x^2+4x-3)(x^2+4x+3) geq 0\\a); ; x^2+4x-3=0; ,\\ D/4=4+3=7,\\x_1=-2-sqrt7approx -4,65; ,; ; ; x_2=-2+sqrt7approx 0,65\\b); ; x^2+4x+3=0\\x_1=-3,; ; x_2=-1; ; (teorema; Vieta)\\c); ; ; (x+2+sqrt7)(x+2-sqrt7)(x+1)(x+3) geq 0\\+++[-2-sqrt7]--[-3, ]+++[-1, ]--[-2+sqrt7, ]++$

$xne0; ,; ; xne -4\\xin (-infty ;-2-sqrt7, ]cup [-3;-1, ]cup [-2+sqrt7;+infty )$