Step 1:
The given inequality is $\large\frac{1}{2} \bigg(\large\frac{3x}{5} +4 \bigg) $$ \geq \large\frac{1}{3} $$(x-6)$
Multiplying by 6 on both sides,
$=> 3 \bigg( \large\frac{3x}{5} +4 \bigg) \geq 2(x-6)$
$=> \large\frac{9x}{5} $$+12 \geq 2(x-6)$
Adding 12 on both sides ,
$\large\frac{9x}{5}$$+12 \geq 2x-12$
Adding $ -\large\frac{9x}{5}$ on both sides
=> $ 24 \geq 2x -\large\frac{9x}{5}$
$=> 24 \geq \large\frac{10x -9x}{5}$
$=> 24 \geq \large\frac{x}{5}$
Multiplying by 5 on both sides,
$120 \geq x$
Step 2:
All real numbers Which are less than or equal to $120$satisfy the given inequality
The solution set is $(-\infty,120]$
Hence B is the correct answer.