Question 38633
Let y = sqrt((x+1)(x+3)/(x+2).)Find all the real values of x for which y takes real
values.
FOR Y TO BE REAL (X+1)(X+3)/(X+2)SHOULD BE GREATER THAN OR EQUAL TO ZERO.
CASE 1......NR AND DR +VE
NR....(X+3)(X+1)SHOULD BE POSITIVE THAT IS 
X>-1........I,OR......X<-3....................II
DR...(X+2) SHOULD BE +VE THAT IS 
X>-2.............III
COMBINING I,II,III,WE GET FOR CASE 1.............X>-1........ANSWER SET 1
CASE 2....NR AND DR -VE.
NR.....(X+3)(X+1) SHOULD BE - VE.
THAT IS -3<X<-1..........................IV
DR....(X+2) SHOULD BE -VE....X<-2.............V
COMBINING IV AND V WE GET -3<X<-2.................................ANSWER SET 2
SINCE NR COULD BE ZERO TOO,WE GET X=-3 AND -1 FOR THIS CASE.HENCE THE TOTAL SOLUTION SET IS
X=-3,X=-1,X>-1,-3<X<-2...OR
X>=-1....AND.....-3<=X<-2