Solve each of the following system of equations in R. 12. x + 5 > 2(x + 1), 2 − x < 3 (x + 2)

Solution:

$x+5>2(x+1)$

$\Rightarrow x+5>2 x+2$

$\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 $(-1,3)$.