The equation $a\sin x + \cos 2x = 2a – 7$ possesses a solution if.

$a\sin x+\cos 2x=2a-7$

$a\sin x+1-2{\sin }^{2}x-2a+7=0$

$2{\sin }^{2}x-a\sin x+2a-8=0$

$\sin x=\frac{a\pm \sqrt{a^2-4\left(2\right)\left(2a-8\right)}}{2\left(2\right)}$

$\sin x=\frac{a\pm \sqrt{{\left(a-8\right)}^2}}{4}$

$\sin x=\frac{a+a-8}{4}=\frac{2a-8}{4}$

$-1\le \frac{a-4}{2}<1$

$-2\le a-4\le 2$

$2\le a\le 6$