Question

Solve the following system of inequations in $R$. $x+5>2(x+1)$,
$2-x<3(x+2)$

Answer 1

Given system of inequations:
$x+5>2(x+1)and2-x<3(x+2)$
Let us consider the first inequation.
$x+5>2(x+1)$
$\Rightarrow x+5>2x+2$
Transposing 5 to R.H.S. and 2x to L.H.S.
$\Rightarrow x-2x>2-5$
$\Rightarrow -x>-3$
$\Rightarrow x<3$
($\therefore$ Inequality sign reverses when both sides of inequation is multiplied by negative number." )] $\therefore xϵ(-\infty ,3)\cdots \left(1\right)$
Now, let us consider the second inequation.
$2-x<3(x+2)$
$\Rightarrow 2-x<3x+6$
Transposing 2 to "R.H.S." and 3𝑥 to L.H.S.
$\Rightarrow -x-3x<6-2$
$\Rightarrow -4x<4$
$\Rightarrow \frac{-4x}{-4}>\frac{4}{-4}$
$(\therefore \text{Inequality sign reverses when both sides of inequation is divided by negative number.})]$
$\therefore x>-1$
$\therefore xϵ(-1,\infty )\dots \left(2\right)$
From (1) and (2) we get
$xϵ(-\infty ,3)\cap (-1,\infty )$
$\therefore xϵ(-1,3)$

Thus, the solution set of given system of inequations is (-1, 3)