(5^(2*x+1)-75*(1/5)^2*x-10)*1/(x+2)

$$frac{1}{x + 2} left(5^{2 x + 1} – 3 x – 10right) leq 0$$

Дано неравенство:
$$frac{1}{x + 2} left(5^{2 x + 1} – 3 x – 10right) leq 0$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$frac{1}{x + 2} left(5^{2 x + 1} – 3 x – 10right) = 0$$
Решаем:
$$x_{1} = – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right)$$
$$x_{2} = – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )},-1 right )} + log{left (95367431640625 right )}right)$$
$$x_{1} = – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right)$$
$$x_{2} = – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )},-1 right )} + log{left (95367431640625 right )}right)$$
Данные корни
$$x_{1} = – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right)$$
$$x_{2} = – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )},-1 right )} + log{left (95367431640625 right )}right)$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} leq x_{1}$$
Возьмём например точку
$$x_{0} = x_{1} – frac{1}{10}$$
=

/ / 3 ___
| | 2*/ 5 ||
| | ——-||
| | 46875 ||
3*LambertW -log5 // + log(95367431640625) 1
– ———————————————— – —
1 10
6*log (5)

=
$$- frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right) – frac{1}{10}$$
подставляем в выражение
$$frac{1}{x + 2} left(5^{2 x + 1} – 3 x – 10right) leq 0$$

/ / / 3 ___
| | | 2*/ 5 || |
| | | ——-|| |
| | | 46875 || | / / / 3 ___
| 3*LambertW -log5 // + log(95367431640625) 1 | | | | 2*/ 5 || |
2*|- ———————————————— – –| + 1 | | | ——-|| |
| 1 10| | | | 46875 || |
6*log (5) / 1 | 3*LambertW -log5 // + log(95367431640625) 1 |
5 – 75*–*|- ———————————————— – –| – 10
2 | 1 10|
5 6*log (5) /
————————————————————————————————————————————— <= 0 1 / / / 3 ___ | | | 2*/ 5 || | | | | -------|| | | | | 46875 || | | 3*LambertW -log5 // + log(95367431640625) 1 | |- ------------------------------------------------ - -- + 2| | 1 10 | 6*log (5) /

/ / 3 ___
| | 2*/ 5 ||
| | ——-|| / / 3 ___
| | 46875 || | | 2*/ 5 ||
4 3*LambertW -log5 // + log(95367431640625) | | ——-||
– – ———————————————— | | 46875 ||
97 5 3*log(5) 3*LambertW -log5 // + log(95367431640625)
– — + 5 + ————————————————
10 2*log(5) <= 0 --------------------------------------------------------------------------------------------------------------- / / 3 ___ | | 2*/ 5 || | | -------|| | | 46875 || 19 3*LambertW -log5 // + log(95367431640625) -- - ------------------------------------------------ 10 6*log(5)

значит одно из решений нашего неравенства будет при:
$$x leq – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right)$$

_____ _____
/
——-•——-•——-
x1 x2

Другие решения неравенства будем получать переходом на следующий полюс
и т.д.
Ответ:
$$x leq – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right)$$
$$x geq – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )},-1 right )} + log{left (95367431640625 right )}right)$$

/ / / / / 3 ___ / / / / 3 ___
| | | | | 2*/ 5 || | | | | | | 2*/ 5 | | | ||
| | | | | ——-|| | | | | | | ——-| | | ||
| | | | | 46875 || | | | | | | 46875 | | | ||
| | -3*LambertW -log5 // + log(95367431640625)/ | | -3*LambertW -log5 /, -1/ + log(95367431640625)/ ||
Or|And|x <= ----------------------------------------------------, -oo < x|, And|x <= --------------------------------------------------------, -2 < x|| 6*log(5) / 6*log(5) //

$$left(x leq – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right) wedge -infty < xright) vee left(x leq - frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )},-1 right )} + log{left (95367431640625 right )}right) wedge -2 < xright)$$

/ / / 3 ___ / / / 3 ___
| | | 2*/ 5 || | | | | 2*/ 5 | | |
| | | ——-|| | | | | ——-| | |
| | | 46875 || | | | | 46875 | | |
-3*LambertW -log5 // + log(95367431640625)/ -3*LambertW -log5 /, -1/ + log(95367431640625)/
(-oo, —————————————————-] U (-2, ——————————————————–]
6*log(5) 6*log(5)

$$x in left(-infty, – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )} right )} + log{left (95367431640625 right )}right)right] cup left(-2, – frac{1}{6 log{left (5 right )}} left(3 {Lambertw}{left (- log{left (5^{frac{2 sqrt[3]{5}}{46875}} right )},-1 right )} + log{left (95367431640625 right )}right)right]$$