Quadratic inequalities - Help!?

a) (2x-5) / (x+1) < 1

note that a square is always positive so if we multiply by a square we don't have to reverse the inequality

Multiply by (x+1)²

(2x-5)(x+1) < (x+1)²

(2x-5)(x+1) - (x+1)² < 0

factor an (x+1)

[(2x-5) - (x+1)](x+1) < 0

(x-6)(x+1) < 0

this is an upright parabola, with roots -1 & 6

so it is negative (ie below the x axis) between the roots

-1<x<6

b) (2x-7) / (x+5) > 4

An alternate way to solve these problems is to just multiply by x+5 then consider 2 cases

case1: assume x+5>0 (ie x>-5)

multiply by x+5, don't reverse inequality

2x-7>4(x+5)

0>2x+27

x<-27/2

this yields no solutions since you can't have both x>-5 and x<-27/2

case2: assume x+5<0 (ie x<-5)

multiply by x+5, reverse inequality since x+5 is negative

2x-7<4(x+5)

0<2x+27

x>-27/2

so we have -27/2 < x < -5

,.,.,