Disequazione fratta $\dfrac{2x+3}{x-4}<1$

Text

Solve the fractional inequality $\frac{2x+3}{x-4}<1.$

Solution

Move everything to the left: $\frac{2x+3}{x-4}-1<0\implies \frac{2x+3-(x-4)}{x-4}<0\implies \frac{x+7}{x-4}<0.$ The ratio is negative between the two key points $x=-7$ and $x=4$ (both excluded): $\boxed{-7<x<4.}$