45*1/((log(x)^2+6*log(x))^2)+14*1/(log(x)^2+6*log(x)*1/log(2))+1>=0

$$frac{45}{left(log^{2}{left (x right )} + 6 log{left (x right )}right)^{2}} + frac{14}{log^{2}{left (x right )} + frac{6 log{left (x right )}}{log{left (2 right )}}} + 1 geq 0$$

Дано неравенство:
$$frac{45}{left(log^{2}{left (x right )} + 6 log{left (x right )}right)^{2}} + frac{14}{log^{2}{left (x right )} + frac{6 log{left (x right )}}{log{left (2 right )}}} + 1 geq 0$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$frac{45}{left(log^{2}{left (x right )} + 6 log{left (x right )}right)^{2}} + frac{14}{log^{2}{left (x right )} + frac{6 log{left (x right )}}{log{left (2 right )}}} + 1 = 0$$
Решаем:
Дано уравнение
$$frac{45}{left(log^{2}{left (x right )} + 6 log{left (x right )}right)^{2}} + frac{14}{log^{2}{left (x right )} + frac{6 log{left (x right )}}{log{left (2 right )}}} + 1 = 0$$
преобразуем
$$frac{left(log^{5}{left (x right )} + 12 log^{4}{left (x right )} + 50 log^{3}{left (x right )} + 168 log^{2}{left (x right )} + log{left (x^{549} right )}right) log{left (2 right )} + 6 log^{4}{left (x right )} + 72 log^{3}{left (x right )} + 216 log^{2}{left (x right )} + 270}{left(left(log^{2}{left (x right )} + 12 log{left (x right )} + 36right) log{left (2 right )} log{left (x right )} + 6 log^{2}{left (x right )} + 72 log{left (x right )} + 216right) log^{2}{left (x right )}} = 0$$
$$frac{left(log^{5}{left (x right )} + 12 log^{4}{left (x right )} + 50 log^{3}{left (x right )} + 168 log^{2}{left (x right )} + log{left (x^{549} right )}right) log{left (2 right )} + 6 log^{4}{left (x right )} + 72 log^{3}{left (x right )} + 216 log^{2}{left (x right )} + 270}{left(log^{5}{left (x right )} + 12 log^{4}{left (x right )} + 36 log^{3}{left (x right )}right) log{left (2 right )} + 6 log^{4}{left (x right )} + 72 log^{3}{left (x right )} + 216 log^{2}{left (x right )}} = 0$$
Сделаем замену
$$w = log{left (2 right )}$$
Дано уравнение:
$$frac{w left(log^{5}{left (x right )} + 12 log^{4}{left (x right )} + 50 log^{3}{left (x right )} + 168 log^{2}{left (x right )} + log{left (x^{549} right )}right) + 6 log^{4}{left (x right )} + 72 log^{3}{left (x right )} + 216 log^{2}{left (x right )} + 270}{w left(log^{5}{left (x right )} + 12 log^{4}{left (x right )} + 36 log^{3}{left (x right )}right) + 6 log^{4}{left (x right )} + 72 log^{3}{left (x right )} + 216 log^{2}{left (x right )}} = 0$$
Домножим обе части ур-ния на знаменатель 6*log(x)^4 + 72*log(x)^3 + 216*log(x)^2 + w*(log(x)^5 + 12*log(x)^4 + 36*log(x)^3)
получим:
$$frac{1}{left(w left(log^{2}{left (x right )} + 12 log{left (x right )} + 36right) log{left (x right )} + 6 log^{2}{left (x right )} + 72 log{left (x right )} + 216right) log^{2}{left (x right )}} left(w left(log^{5}{left (x right )} + 12 log^{4}{left (x right )} + 36 log^{3}{left (x right )}right) + 6 log^{4}{left (x right )} + 72 log^{3}{left (x right )} + 216 log^{2}{left (x right )}right) left(w left(log^{5}{left (x right )} + 12 log^{4}{left (x right )} + 50 log^{3}{left (x right )} + 168 log^{2}{left (x right )} + log{left (x^{549} right )}right) + 6 log^{4}{left (x right )} + 72 log^{3}{left (x right )} + 216 log^{2}{left (x right )} + 270right) = 0$$
Раскрываем скобочки в левой части ур-ния

6*log+x^4 + 72*logx^3 + 216*logx^2 + wlog+x^5 + 12*logx^4 + 36*logx^3))270+6*log+x^4 + 72*logx^3 + 216*logx^2 + wlog+x^5 + 12*logx^4 + 50*logx^3 + 168*logx^2 + logx+549))/216+/6*log+/x^2 + 72*logx + w36+log+x^2 + 12*logx)*logx)*logx^2) = 0

Разделим обе части ур-ния на (6*log(x)^4 + 72*log(x)^3 + 216*log(x)^2 + w*(log(x)^5 + 12*log(x)^4 + 36*log(x)^3))*(270 + 6*log(x)^4 + 72*log(x)^3 + 216*log(x)^2 + w*(log(x)^5 + 12*log(x)^4 + 50*log(x)^3 + 168*log(x)^2 + log(x^549)))/(w*(216 + 6*log(x)^2 + 72*log(x) + w*(36 + log(x)^2 + 12*log(x))*log(x))*log(x)^2)

w = 0 / ((6*log(x)^4 + 72*log(x)^3 + 216*log(x)^2 + w*(log(x)^5 + 12*log(x)^4 + 36*log(x)^3))*(270 + 6*log(x)^4 + 72*log(x)^3 + 216*log(x)^2 + w*(log(x)^5 + 12*log(x)^4 + 50*log(x)^3 + 168*log(x)^2 + log(x^549)))/(w*(216 + 6*log(x)^2 + 72*log(x) + w*(36 + log(x)^2 + 12*log(x))*log(x))*log(x)^2))

Получим ответ: w = -(270 + 6*log(x)^4 + 72*log(x)^3 + 216*log(x)^2)/(log(x)^5 + 12*log(x)^4 + 50*log(x)^3 + 168*log(x)^2 + log(x^549))
делаем обратную замену
$$log{left (2 right )} = w$$
подставляем w:
$$x_{1} = 1$$
$$x_{2} = 0.295779216289 – 0.42595480358 i$$
$$x_{3} = 0.295779216289 + 0.42595480358 i$$
Исключаем комплексные решения:
$$x_{1} = 1$$
Данные корни
$$x_{1} = 1$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} leq x_{1}$$
Возьмём например точку
$$x_{0} = x_{1} – frac{1}{10}$$
=
$$0.9$$
=
$$0.9$$
подставляем в выражение
$$frac{45}{left(log^{2}{left (x right )} + 6 log{left (x right )}right)^{2}} + frac{14}{log^{2}{left (x right )} + frac{6 log{left (x right )}}{log{left (2 right )}}} + 1 geq 0$$
$$1 + frac{14}{frac{6 log{left (0.9 right )}}{log{left (2 right )}} + log^{2}{left (0.9 right )}} + frac{45}{left(6 log{left (0.9 right )} + log^{2}{left (0.9 right )}right)^{2}} geq 0$$

14
117.66544478731 + ————————————–
0.632163093946958 >= 0
0.0111008382596831 – —————–
log(2)

значит решение неравенства будет при:
$$x leq 1$$

_____
——-•——-
x1

$$left(-infty < x wedge x < e^{- frac{6}{log{left (2 right )}}}right) vee left(1 < x wedge x < inftyright) vee left(x < 1 wedge e^{-6} < xright) vee left(x < e^{-6} wedge e^{- frac{6}{log{left (2 right )}}} < xright)$$

-6 -6
—— ——
log(2) log(2) -6 -6
(-oo, e ) U (e , e ) U (e , 1) U (1, oo)

$$x in left(-infty, e^{- frac{6}{log{left (2 right )}}}right) cup left(e^{- frac{6}{log{left (2 right )}}}, e^{-6}right) cup left(e^{-6}, 1right) cup left(1, inftyright)$$