Question

Solve the following inequality

$\frac{-2x+5}{x+6}>-2$

Answer 1

$\frac{-2x+5}{x+6}>-2$

$\frac{-2x+5}{x+6}+2>0$

$\frac{-2x+5+2x+12}{x+6}>0$

$\therefore \frac{17}{x+6}>0$

The above equation represents

that $\frac{17}{x+6}>0$ must be positive i.e $>0$

for $\frac{17}{x+6}>0,x+6$ must be greater

$x+6>0$

$\therefore x>-\mathrm{6...}\left(1\right)$

Also $\frac{-2x+5}{x+6}>-2$

$\frac{+2x+6}{-2x+5}<\frac{-1}{2}$ [Taking reciprocal on both side changes the sign of inequality]

$\frac{x+6}{-2+5}+\frac{1}{2}<0$

$\frac{2\times 12+(-2\times +5)}{-2x+5}<0$

$\frac{17}{-2x+5}<0$

For the above equation to be true

$-2x+5<0$

$x>\frac{5}{2}....\left(2\right)$

After combining both the solutions

the required solution we get is

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