Question

Solve the following inequality:
$|x+2|-|x-1|<x-\frac{3}{2}.$

Answer 1

there are 3 cases they are

$i.)x<-2,ii.)-2<x<1,iii.)x>1$

the respective equations are

$i.)-2-x-(1-x)<x-3/2,ii.)x+2-(1-x)<x-\frac{3}{2},iii.)x+2-(x-1)<x-3/2$

in first case

$i.)x>-3/2$ and we assumed $x<-2$

therefore there is no solution in this case

in second case

$x<-5/2$ and we assumed $x>-2$

therefore there is no solution in this case

in third case

$x>9/2$ and we assumed $x>1$

therefore the solution in this case is $x>\frac{9}{2}$

the solution is union of all solutions in all cases

which is $x\in (\frac{9}{2},\infty )$