Answer
$(6,∞)$
Work Step by Step
$2x-3 \lt 3x+1 \lt 4x-5$
Using the inequality properties, add $-1$ to all parts.
$2x-3-1 \lt 3x+1 -1 \lt 4x-5-1$
$2x-4 \lt 3x \lt 4x-6$
Add $-2x$ to all parts.
$2x-4-2x \lt 3x-2x \lt 4x-6-2x$
$-4 \lt x \lt 2x-6$
We can write it as
$-4 \lt x $ and $ x \lt 2x-6$
Again apply the inequality properties at right hand side only.
Add $-2x$
$-4 \lt x $ and $ x -2x \lt 2x-6-2x$
$-4 \lt x $ and $-x \lt -6$
Divide by $-1$ and reverse the inequality symbol.
$-4 \lt x $ and $\frac{-x}{-1} \gt\frac{-6}{-1}$
$-4 \lt x $ and $x \gt 6$
In interval notation: $(6,∞)$
