Question

Solve the given inequality for real
$x:\frac{2x-1}{3}\ge \frac{3x-2}{4}-\frac{2-x}{5}$

Answer 1

Given, $\frac{2x-1}{3}\ge \frac{3x-2}{4}-\frac{2-x}{5}$
$\Rightarrow \frac{2x-1}{3}\ge \frac{5(3x-2)-4(2-x)}{20}$
$\Rightarrow \frac{2x-1}{3}\ge \frac{15x-10-8+4x}{20}$

$\Rightarrow 20(2x-1)\ge 3(19x-18)$

$\Rightarrow 40x-20\ge 57x-54$

$\Rightarrow 40x-57x\ge 20-54$


$\Rightarrow -17x\ge -34$

$\Rightarrow -x\ge \frac{-34}{17}$

$\Rightarrow -x\ge -2$

Multiply both sides by $-1$ also reverse the inequality signs.
$\Rightarrow (-1)\times (-x)\le (-1)\times (-2)$
$\Rightarrow x\le 2$

Since $x$ is a real number which is less than or equal to $2$.
The solution set is $x\in (-\infty ,2]$