Question

Решите систему неравенств
$left { {{frac{1}{(x+4)}-frac{1}{(2x+4)} textless =0} atop {x^2+5x+4 textless =0} right.$

Answer 1

$left{ begin{gathered} frac{1}{x+4} - dfrac{1}{2x+4} le 0 \ x^{2} + 5x + 4 le 0 \ end{gathered} right.$

$ODZ: left{ begin{gathered} x + 4 ne 0 \ 2x + 4 ne 0 \ end{gathered} right. ; left{ begin{gathered} x ne -4 \ x ne -2 \ end{gathered} right.$

$left{ begin{gathered} frac{2x+4 - x - 4}{(2x+4)(x+4)} le 0 \ x^{2} + 5x + 4 le 0 \ end{gathered} right.$

$left{ begin{gathered} frac{x}{2(x+2)(x+4)} le 0 (1) \ (x+1)(x+4) le 0 (2) \ end{gathered} right.$

$left{ begin{gathered} x in (-infty ; -4)cup (-2;0] \ x in [-4;-1] \ end{gathered} right.$

Пересечем множество решений и учтём ОДЗ (3):

$x in (-2;-1]$

Ответ: x ∈ (-2;-1]