Question 32245
PROVE ALGEBRAICALLY THAT: ( (xy) / (1+(xz)) ) > ( ((xy)+(ab)) / (1 + (xz) + (ac)) ) REDUCES TO: (b/c) < ( (xy) / (1+(xz)) ). 
I HAVE NO IDEA WHERE TO EVEN START!!
DONT GET SCARED!KEEP UP THE SPIRIT.WE SHOULD LEARN HOWTO REMOVE THE VEILS OR CAMOUFLAGE TO LOOK AT THE REAL FRUIT.
FOR THIS PURPOSE WE USE ONION PEEL APPROACH...REMOVE THE OUTR LAYERS ONE BY ONE...AND BUILDING BLOCKS..WE TRY TO LEARN FROM SMALL THINGS TO BIG THINGS.
HERE THERE ARE BIG EXPRESSIONS..LET US SIMPLIFY BY NEW SYMBOLS.
LET US CALL L= ( (xy) / (1+(xz)) )......AND M = ( ((xy)+(ab)) / (1 + (xz) + (ac))
AND K = (b/c).....HENCE GIVEN L > M , TST.....K < L..........
LET US STUDY SOME EXAMPLES...SUPPOSE 

IF L=4/2=NUMBER GREATER THAN 1...THEN YOU CAN EASILY SEE THAT ADDING SAME NUMBER...SAY 2 TO NR AND DR WILL MAKE THE NEW NUMBER (4+2)/(2+2)=6/4=3/2 SMALLER.
  
IF L=4/2=NUMBER GREATER THAN 1...THEN YOU CAN EASILY SEE THAT SUBTRACTINGING SAME NUMBER...SAY 1 FROM NR AND DR WILL MAKE THE NEW NUMBER (4-1)/(2-1)=3/1=3 BIGGER.

IF L=2/4=NUMBER SMALLER THAN 1...THEN YOU CAN EASILY SEE THAT ADDING SAME NUMBER...SAY 2 TO NR AND DR WILL MAKE THE NEW NUMBER (2+2)/(4+2)=4/6=2/3 BIGGER.

IF L=2/4=NUMBER SMALLER THAN 1...THEN YOU CAN EASILY SEE THAT SUBTRACTING SAME NUMBER...SAY 1 FROM NR AND DR WILL MAKE THE NEW NUMBER (2-1)/(4-1)=1/3= SMALLER.
  THIS IS FUNDAMENTAL PROPERTY OF FRACTIONS. 
    NOW WE HAVE 
K=B/C..OR..B=KC....SO AB =KAC
WE ARE GIVEN THAT
L>(L+KAC)/(1+AC).....FROM THIS AND ABOVE PROPERTIES OF FRACTIONS,WE SEE THAT 
IF L >1...THEN AC IS POSITIVE...AND IF L< 1 THEN AC IS NEGATIVE.
I SHALL SHOW YOU THE PROOF FOR THE FIRST ALTERNATIVE AND YOU CAN DO FOR THE SECOND ALTERNATIVE.
L >1 AND AC IS POSITIVE
 L-(L+KAC)/(1+AC)>0
(L+LAC-L-KAC)/(1+AC)>0
AC(L-K)>0...SINCE AC IS POSITIVE AND HENCE 1+AC IS POSITIVE.
L-K>0........SINCE AC IS POSITIVE
L>K...OR K<L...PROVED.