Elementary system of inequalities - set of solutions

$3x-1\le 1+4x$

$\frac{5}{6}x+\frac{2}{3}\ge 1+\frac{1}{2}x$

$\frac{x-1}{6}<\frac{1}{4}$

$\Rightarrow $

$x\ge -2$

$x\ge 1$

$x<\frac{5}{2}$

I compared the above reduced inequalities which gave me as answer that the solutions are from [$\frac{5}{2}$]. I think it is correct but I'm not sure. Can someone affirm this or debunk this please and what's the reason?

$3x-1\le 1+4x$

$\frac{5}{6}x+\frac{2}{3}\ge 1+\frac{1}{2}x$

$\frac{x-1}{6}<\frac{1}{4}$

$\Rightarrow $

$x\ge -2$

$x\ge 1$

$x<\frac{5}{2}$

I compared the above reduced inequalities which gave me as answer that the solutions are from [$\frac{5}{2}$]. I think it is correct but I'm not sure. Can someone affirm this or debunk this please and what's the reason?