Solve each of the following system of equations in R.

Solution:

$5 x-7<3(x+3)$

$\Rightarrow 5 x-7<3 x+9$

$\Rightarrow 5 x-3 x<9+7$

$\Rightarrow 2 x<16$

$\Rightarrow x<8$

$\Rightarrow x \in(-\infty, 8) \quad \ldots(\mathrm{i})$

Also, $1-\frac{3 x}{2} \geq x-4$

$\Rightarrow x-4 \leq 1-\frac{3 x}{2}$

$\Rightarrow x+\frac{3 x}{2} \leq 1+4$

$\Rightarrow \frac{2 x+3 x}{2} \leq 5$

$\Rightarrow 5 x \leq 10$

$\Rightarrow x \leq 2$

$\Rightarrow x \in(-\infty, 2] \quad \ldots$ (ii)

Hence, the solution of the given set of inequalities is the intersection of (i) and (ii).

$(-\infty, 8) \cap(-\infty, 2]=(-\infty, 2]$

Hence, the solution of the given set of inequalities is $(-\infty, 2]$.