Question

Solve the following system of inequations in $R$.
$2(x-6)<3x-7$,
$11-2x<6-x$

Answer 1

Given system of inequations:
$2(x-6)<3x-7and11-2x<6-x$
Let us consider the first inequation.
$2(x-6)<3x-7$
$\Rightarrow 2x-12<3x-7$
Transposing −12 to R.H.S. and 3x to L.H.S.
$\Rightarrow 2x-3x<-7+12$
$\Rightarrow -x<5$
$\Rightarrow x>-5$
$[\therefore \text{Inequality sign reverses when both sides of inequation is multiplied by negative number.}]$
$\therefore xϵ(-5,\infty )\dots \left(1\right)$

Now, let us consider the second inequation.
$11-2x<6-x$
Transposing 11 to R.H.S. and -x to L.H.S.
$\Rightarrow -2x+x<6-11$
$\Rightarrow -x<-5$
$\Rightarrow x>5$
$[\therefore \text{Inequality sign reverses when both sides of inequation is multiplied by negative number.}]$
$\therefore xϵ(5,\infty )\dots \left(2\right)$
From (1) and (2)we get,
$xϵ(-5,\infty )\cap (5,\infty )$
$\therefore xϵ(5,\infty )$

Thus, the solution set of the given system of inequations is $(5,\infty )$