sqrt((x-log(3)*1/log(2))*(log(6)*1/log(7)-x))>=0

$$sqrt{left(- x + frac{log{left (6 right )}}{log{left (7 right )}}right) left(x – frac{log{left (3 right )}}{log{left (2 right )}}right)} geq 0$$

Дано неравенство:
$$sqrt{left(- x + frac{log{left (6 right )}}{log{left (7 right )}}right) left(x – frac{log{left (3 right )}}{log{left (2 right )}}right)} geq 0$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$sqrt{left(- x + frac{log{left (6 right )}}{log{left (7 right )}}right) left(x – frac{log{left (3 right )}}{log{left (2 right )}}right)} = 0$$
Решаем:
$$sqrt{left(- x + frac{log{left (6 right )}}{log{left (7 right )}}right) left(x – frac{log{left (3 right )}}{log{left (2 right )}}right)} = 0$$
преобразуем
$$- x^{2} + x left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right) – frac{log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} = 0$$
Раскроем выражение в уравнении
$$- x^{2} + x left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right) – frac{log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} = 0$$
Получаем квадратное уравнение
$$- x^{2} + frac{x log{left (6 right )}}{log{left (7 right )}} + frac{x log{left (3 right )}}{log{left (2 right )}} – frac{log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} = 0$$
Это уравнение вида

a*x^2 + b*x + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$x_{1} = frac{sqrt{D} – b}{2 a}$$
$$x_{2} = frac{- sqrt{D} – b}{2 a}$$
где D = b^2 – 4*a*c – это дискриминант.
Т.к.
$$a = -1$$
$$b = frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}$$
$$c = – frac{log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}}$$
, то

D = b^2 – 4 * a * c =

(log(3)/log(2) + log(6)/log(7))^2 – 4 * (-1) * (-log(3)*log(6)/(log(2)*log(7))) = (log(3)/log(2) + log(6)/log(7))^2 – 4*log(3)*log(6)/(log(2)*log(7))

Т.к. D > 0, то уравнение имеет два корня.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b – sqrt(D)) / (2*a)

или
$$x_{1} = – frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
$$x_{2} = frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
$$x_{1} = – frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
$$x_{2} = frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
$$x_{1} = – frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
$$x_{2} = frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
Данные корни
$$x_{1} = – frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
$$x_{2} = frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} leq x_{1}$$
Возьмём например точку
$$x_{0} = x_{1} – frac{1}{10}$$
=

________________________________________
/ 2
/ / log(3) log(6) 4*log(3)*log(6)
/ |——- + ——-| – —————
/ | 1 1 | 1 1
/ log (2) log (7)/ log (2)*log (7) log(3) log(6) 1
– ———————————————- + ——— + ——— – —
2 1 1 10
2*log (2) 2*log (7)

=
$$- frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} – frac{1}{10} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
подставляем в выражение
$$sqrt{left(- x + frac{log{left (6 right )}}{log{left (7 right )}}right) left(x – frac{log{left (3 right )}}{log{left (2 right )}}right)} geq 0$$

_____________________________________________________________________________________________________________________________________________________________________________________
/ / ________________________________________ / ________________________________________
/ | / 2 | | / 2 |
/ | / / log(3) log(6) 4*log(3)*log(6) | | / / log(3) log(6) 4*log(3)*log(6) |
/ | / |——- + ——-| – ————— | | / |——- + ——-| – ————— |
/ | / | 1 1 | 1 1 | | / | 1 1 | 1 1 |
/ | / log (2) log (7)/ log (2)*log (7) log(3) log(6) 1 log(3)| | log(6) / log (2) log (7)/ log (2)*log (7) log(3) log(6) 1 |
/ |- ———————————————- + ——— + ——— – — – ——-|*|——- – – ———————————————- + ——— + ——— – –| >= 0
/ | 2 1 1 10 1 | | 1 2 1 1 10|
/ 2*log (2) 2*log (7) log (2)/ log (7) 2*log (2) 2*log (7) /

_____________________________________________________________________________________________________________________________________________________
/ / ______________________________________ / ______________________________________
/ | / 2 | | / 2 |
/ | / /log(3) log(6) 4*log(3)*log(6) | | / /log(3) log(6) 4*log(3)*log(6) |
/ | / |—— + ——| – ————— | | / |—— + ——| – ————— | >= 0
/ | 1 / log(2) log(7)/ log(2)*log(7) log(6) log(3) | |1 / log(2) log(7)/ log(2)*log(7) log(6) log(3) |
/ |- — – ——————————————- + ——– – ——–|*|– + ——————————————- + ——– – ——–|
/ 10 2 2*log(7) 2*log(2)/ 10 2 2*log(7) 2*log(2)/

Тогда
$$x leq – frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$
не выполняется
значит одно из решений нашего неравенства будет при:
$$x geq – frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}} wedge x leq frac{1}{2} sqrt{- frac{4 log{left (3 right )} log{left (6 right )}}{log{left (2 right )} log{left (7 right )}} + left(frac{log{left (6 right )}}{log{left (7 right )}} + frac{log{left (3 right )}}{log{left (2 right )}}right)^{2}} + frac{log{left (6 right )}}{2 log{left (7 right )}} + frac{log{left (3 right )}}{2 log{left (2 right )}}$$

_____
/
——-•——-•——-
x1 x2