Solve each of the following system of equations in R.
12. x + 5 > 2(x + 1), 2 − x < 3 (x + 2)
$x+5>2(x+1)$
$\Rightarrow x+5>2 x+2$
$\Rightarrow 2 x+2 $\Rightarrow 2 x-x<5-2$ $\Rightarrow x<3$ $\begin{array}{ll}\Rightarrow x \in(-\infty, 3) & \ldots(\mathrm{i})\end{array}$ Also, $2-x<3(x+2)$ $\Rightarrow 2-x<3 x+6$ $\Rightarrow 3 x+6>2-x$ $\Rightarrow 3 x+x>2-6$ $\Rightarrow 4 x>-4$ $\Rightarrow x>-1$ $\Rightarrow x \in(-1, \infty) \quad \ldots$ (ii) Hence, the solution of the given set of inequations is the intersection of (i) and (ii). $(-\infty, 3) \cap(-1, \infty)=(-1,3)$ Thus, the solution of the given set of inequations is $(-1,3)$.
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