To solve the linear inequality $4b - 3(3 - b) > 5(b - 6) + 2b$, systematically simplify both sides heavily by distributing, extracting $4b - 9 + 3b > 5b - 30 + 2b$. Combining corresponding like terms accurately returns $7b - 9 > 7b - 30$. Subtracting $7b$ logically from both sides completely zeroes out the variable terms and leaves only $-9 > -30$. Since this denotes an inherently true mathematical statement irrespective of variables, the unique inequality robustly functions as an identity boasting an infinite solution of all real numbers. The entire number line is permanently shaded for accurate geometric representation, natively defining $(-\infty, \infty)$ in standard interval notation.