$a)\left(x+3\right)\left(x-3\right)+2\le {\left(x-4\right)}^2-3\left(1-x\right)$  

$x^2-9+2\le {\left(x-4\right)}^2-3+3x$  

$x^2-7\le x^2-8x+16-3+3x$  

$-7\le -5x+13$  

$5x\le 20$  

$x\le 4$  

$x\in \left(-\infty ,4\right]$  

 

$b)\frac{2x-1}{2}-\frac{x-8}{3}<x\mid \cdot 6$   

$3\left(2x-1\right)-2\left(x-8\right)<6x$  

$6x-3-2x+16<6x$  

$4x+13<6x$  

$13<2x$  

$\frac{13}{2}<x$  

$x\in \left(\frac{13}{2},\infty \right)$  

 

$c)\frac{2}{3}\left(x-2\right)-\frac{3}{4}x>\frac{x-1}{6}+x+4\mid \cdot 12$  

$8\left(x-2\right)-9x>2\left(x-1\right)+12x+48$  

$8x-16-9x>2x-2+12x+48$  

$-x-16>14x+46$  

$-62>15x$  

$-\frac{62}{15}>x$   

$x\in \left(-\infty ,-\frac{64}{15}\right)$  

 

$d)\left(x-\sqrt{5}\right)\left(x+\sqrt{5}\right)-\frac{1}{2}\left(2-x^2\right)\ge -x+\frac{3{\left(x-1\right)}^2}{2}$  

$x^2-5-1+\frac{1}{2}{x}^2\ge -x+\frac{3}{2}{\left(x-1\right)}^2$  

$\frac{3}{2}{x}^2+x-6\ge \frac{3}{2}{\left(x-1\right)}^2$   

$\frac{3}{2}{x}^2+x-6\ge \frac{3}{2}\left(x^2-2x+1\right)$  

$\frac{3}{2}{x}^2+x-6\ge \frac{3}{2}{x}^2-3x+\frac{3}{2}$  

$x-6\ge -3x+\frac{3}{2}$  

$4x\ge \frac{15}{2}$  

$x\ge \frac{15}{8}$  

$x\in \left[\frac{15}{8},\infty \right)$