Дано

$$frac{log{left (left(2 x + 3right)^{3} right )}}{log{left (10 right )}} + frac{2}{log{left (left(2 x + 3right)^{frac{31}{10}} right )}} – 3 < 0$$
Подробное решение
Дано неравенство:
$$frac{log{left (left(2 x + 3right)^{3} right )}}{log{left (10 right )}} + frac{2}{log{left (left(2 x + 3right)^{frac{31}{10}} right )}} – 3 < 0$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$frac{log{left (left(2 x + 3right)^{3} right )}}{log{left (10 right )}} + frac{2}{log{left (left(2 x + 3right)^{frac{31}{10}} right )}} – 3 = 0$$
Решаем:
$$x_{1} = – frac{3}{2} + frac{sqrt{10}}{2} e^{frac{sqrt{93}}{186} sqrt{-80 + 93 log{left (10 right )}} sqrt{log{left (10 right )}}}$$
$$x_{2} = – frac{3}{2} + frac{sqrt{10}}{2 e^{frac{sqrt{93}}{186} sqrt{-80 + log{left (1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 right )}} sqrt{log{left (10 right )}}}}$$
$$x_{1} = – frac{3}{2} + frac{sqrt{10}}{2} e^{frac{sqrt{93}}{186} sqrt{-80 + 93 log{left (10 right )}} sqrt{log{left (10 right )}}}$$
$$x_{2} = – frac{3}{2} + frac{sqrt{10}}{2 e^{frac{sqrt{93}}{186} sqrt{-80 + log{left (1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 right )}} sqrt{log{left (10 right )}}}}$$
Данные корни
$$x_{2} = – frac{3}{2} + frac{sqrt{10}}{2 e^{frac{sqrt{93}}{186} sqrt{-80 + log{left (1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 right )}} sqrt{log{left (10 right )}}}}$$
$$x_{1} = – frac{3}{2} + frac{sqrt{10}}{2} e^{frac{sqrt{93}}{186} sqrt{-80 + 93 log{left (10 right )}} sqrt{log{left (10 right )}}}$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} < x_{2}$$
Возьмём например точку
$$x_{0} = x_{2} – frac{1}{10}$$
=

____ ___________________________________________________________________________________________________________ _________
-/ 93 */ -80 + log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) */ log(10)
———————————————————————————————————————————-
____ 186
3 / 10 *e 1
– – + —————————————————————————————————————————————— – —
2 2 10

=
$$- frac{8}{5} + frac{sqrt{10}}{2 e^{frac{sqrt{93}}{186} sqrt{-80 + log{left (1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 right )}} sqrt{log{left (10 right )}}}}$$
подставляем в выражение
$$frac{log{left (left(2 x + 3right)^{3} right )}}{log{left (10 right )}} + frac{2}{log{left (left(2 x + 3right)^{frac{31}{10}} right )}} – 3 < 0$$

/ 3
|/ / ____ ___________________________________________________________________________________________________________ _________ |
|| | -/ 93 */ -80 + log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) */ log(10) | | |
|| | ———————————————————————————————————————————- | | |
|| | ____ 186 | | |
|| | 3 / 10 *e 1 | | |
log||2*|- – + —————————————————————————————————————————————— – –| + 3| |
2 2 10/ / / 2
——————————————————————————————————————————————————————— + ———————————————————————————————————————————————————————– – 3 < 0 1 / 31 log (10) | --| | 10| |/ / ____ ___________________________________________________________________________________________________________ _________ | || | -/ 93 */ -80 + log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) */ log(10) | | | || | ---------------------------------------------------------------------------------------------------------------------------------- | | | || | ____ 186 | | | 1|| | 3 / 10 *e 1 | | | log ||2*|- - + ------------------------------------------------------------------------------------------------------------------------------------------ - --| + 3| | 2 2 10/ / /

/ 3
|/ ____ ___________________________________________________________________________________________________________ _________ |
|| -/ 93 */ -80 + log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) */ log(10) | |
|| ———————————————————————————————————————————-| |
|| 1 ____ 186 | |
log||- – + / 10 *e | |
2 5 / /
-3 + ——————————————————————————————————————————————————— + ——————————————————————————————————————————————————–
/ 31 log(10) < 0 | --| | 10| |/ ____ ___________________________________________________________________________________________________________ _________ | || -/ 93 */ -80 + log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) */ log(10) | | || ----------------------------------------------------------------------------------------------------------------------------------| | || 1 ____ 186 | | log||- - + / 10 *e | | 5 / /

но

/ 3
|/ ____ ___________________________________________________________________________________________________________ _________ |
|| -/ 93 */ -80 + log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) */ log(10) | |
|| ———————————————————————————————————————————-| |
|| 1 ____ 186 | |
log||- – + / 10 *e | |
2 5 / /
-3 + ——————————————————————————————————————————————————— + ——————————————————————————————————————————————————–
/ 31 log(10) > 0
| –|
| 10|
|/ ____ ___________________________________________________________________________________________________________ _________ |
|| -/ 93 */ -80 + log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) */ log(10) | |
|| ———————————————————————————————————————————-| |
|| 1 ____ 186 | |
log||- – + / 10 *e | |
5 / /

Тогда
$$x < - frac{3}{2} + frac{sqrt{10}}{2 e^{frac{sqrt{93}}{186} sqrt{-80 + log{left (1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 right )}} sqrt{log{left (10 right )}}}}$$
не выполняется
значит одно из решений нашего неравенства будет при:
$$x > – frac{3}{2} + frac{sqrt{10}}{2 e^{frac{sqrt{93}}{186} sqrt{-80 + log{left (1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 right )}} sqrt{log{left (10 right )}}}} wedge x < - frac{3}{2} + frac{sqrt{10}}{2} e^{frac{sqrt{93}}{186} sqrt{-80 + 93 log{left (10 right )}} sqrt{log{left (10 right )}}}$$

_____
/
——-ο——-ο——-
x2 x1

   
4.99
ValeriaSova
Имею два высших международных образования. Опыт написания студенческих и школьных работ более 5 лет. Работаю на трех языках (русский, английский, украинский), пишу курсовые и дипломные работы, рефераты, доклады, контрольные и прочее.