Question

Решите систему неравенств.​ Полное решение

Answer 1

$\displaystyle\bf\\\left \{ {{2x - 7 \leqslant 0} \atop { - 2 {x}^{2} - 7x + 9 < 0 }} \right. \\ \\ 1) \: 2x - 7 \leqslant 0 \\ 2x \leqslant 7 \: \: | \div 2\\ x \leqslant 3.5 \\ \\ 2) \: - 2 {x}^{2} - 7x + 9 < 0 \\ 2 {x}^{2} + 7x - 9 > 0 \\ \\ 2 {x}^{2} + 7x - 9 = 0 \\ a = 2\\ b =7 \\ c = - 9\\ D = {b}^{2} - 4ac = {7}^{2} - 4 \times 2 \times ( - 9) = \\ = 49 + 72 = 121 \\ x_{1} = \frac{ - 7 - 11}{2 \times 2} = - \frac{18}{4} = - 4.5\\ x_{2} = \frac{ - 7 + 11}{2 \times 2} = \frac{4}{4} = 1 \\ \\ {ax}^{2} + bx + c = a(x - x_{1})(x - x_{2}) \\ 2 {x}^{2} + 7x - 9 = 2(x + 4.5)(x - 1) > 0 \\ + + + ( - 4.5) - - - (1) + + + \\ x < - 4.5 \: \: \: and \: \: \: x > 1 \\ \displaystyle\bf\\3) \: \left \{ {{x \leqslant 3.5} \atop {x < - 4.5 \: \: \: and \: \: \: x > 1 }} \right. \\ \\ x \: \epsilon \: ( - \propto; \: - 4.5)U(1; \: 3.5]$