Solve the linear inequation and represent the solution set on a number line:
$-3 \left(x - 7\right) \geq 15 - 7x > \left(\dfrac{x + 1}{3}\right), \;\; x \in R$
Consider $\;$ $-3 \left(x - 7\right) \geq 15 - 7x$
i.e. $\;$ $-3x + 21 \geq 15 - 7x$
i.e. $\;$ $4x \geq -6$
i.e. $\;$ $x \geq -\dfrac{3}{2}$ $\implies$ $- \dfrac{3}{2} \leq x$ $\;\;\; \cdots \; (1)$
Consider $\;$ $15 - 7x > \dfrac{x + 1}{3}$
i.e. $\;$ $45 - 21x > x + 1$
i.e. $\;$ $44 > 22 x$
i.e. $\;$ $2 > x$ $\implies$ $x < 2$ $\;\;\; \cdots \; (2)$
$\therefore \;$ We have from equations $(1)$ and $(2)$, $\;\;$ $- \dfrac{3}{2} \leq x < 2$
$\therefore \;$ The solution set of the given inequation is: $\;$ $\left\{x \mid -\dfrac{3}{2} \leq x < 2, \; x \in R \right\}$
