$a)$

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

$\frac{x^2-9}{3}-\frac{x^2-8x+16}{2}$

$2\left(x^2-9\right)-3\left(x^2-8x+16\right)$

$2x^2-18-3x^2+24x-48$

$-x^2+24x-66$

$24x-66<0\mid +66$

$24x<66\mid :24$

$x<\frac{66}{24}$

$x<\frac{11}{4}$

 

 

 

$b)$

${\left(x-2\right)}^2+1>\frac{x-8}{3}-\left(1-x\right)\left(1+x\right)$

$x^2-4x+4+1>\frac{x-8}{3}-\left(1-x^2\right)$

$x^2-4x+5>\frac{x-8}{3}-1+x^2\mid -x^2$

$-4x+5>\frac{x-8}{3}-1\mid \cdot 3$

$-12x+15>x-8-3$

$-12x+15>x-11\mid -x$

$-13x+15>-11\mid -15$

$-13x>-26\mid :\left(-13\right)$

$x<2$

 

 

$c)$

$1-{\left(3x-2\right)}^2\ge \frac{x-36}{3}-9\left(x+1\right)\left(x-1\right)$

$1-\left(9x^2-12x+4\right)\ge \frac{x-36}{3}-9\left(x^2-1\right)$

$1-9x^2+12x-4\ge \frac{x-36}{3}-9x^2+9\mid +9x^2$

$12x-3\ge \frac{x-36}{3}+9\mid \cdot 3$

$36x-9\ge x-36+27$

$36x-9\ge x-9\mid +9$

$36x\ge x\mid -x$

$35x\ge 0\mid :35$

$x\ge 0$

 

 

 

$d)$

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

$x^2+4x+4-\frac{2x-1}{3}\ge x^2-2-\frac{1}{3}\mid -x^2$

$4x+4-\frac{2x-1}{3}\ge -2-\frac{1}{3}\mid \cdot 3$

$12x+12-\left(2x-1\right)\ge -6-1$

$12x+12-2x+1\ge -7$

$10x+13\ge -7\mid -13$

$10x\ge -20\mid :10$

$x\ge -2$