QUESTIONS- HOPE - INEQUALITIES,RATIO-PROPORTION

QUESTIONS - ANSWERS - HOPE - INEQUALITIES-RATIO & PROPORTION

Hope has left a new comment on your post "QUESTIONS - COMPLEX ANALYSIS":

Solve the following polynomial inequality and give the answer in interval notation. 2x^3 + 4.5x^2 < 0
X^2(2X+4.5)<0
X^2 BEING A PERFECT SQUARE IS ALWAYS >=0.
HENCE
2X+4.5 <0
2X<-4.5
X<-4.5/2 = - 2.25…..

HENCE IN INTERVAL NOTATION
X SHOULD LIE IN (-INFINITY, - 2.25)



What kind of variation is demonstrated by the following function? f(x) = (x / 100) + 1
STRICTLY SPEAKING F(X) DOES NOT FALL UNDER CATEGORY OF “ VARIATION BY DIRECT OR INDIRECT PROPORTION”, BUT
F(X)-1 VARIES DIRECTLY AS X WITH A CONSTANT OF PROPORTION EQUAL TO 1/100.THIS IS BECAUSE ..
X=………………......…100…………… 200……………300….ETC
F(X)-1=(X/100)……1………………..2………………3…ETC.THAT IS
[F(X)-1]/X = CONSTANT ALWAYS.(=1/100 HERE) SO WE SAY
F(X)-1 VARIES DIRECTLY AS X .
BUT NOTE THAT IF WE TAKE ONLY F(X)
X=…………………………......100……………200………………300
F(X) =(X/100)+1…………2…………………3……………….4…ETC
AND F(X)/X = 2/100,3/100,4/100 ETC..THAT IS VARIABLE..NOT CONSTANT.
BUT IS YOU ARE TAUGHT TO CALL 1/100 AS CONSTANT OF PROPORTIONALITY ..THEN YOU MAY CALL SO.



For the constant of proportionality in the following function I get 100. Is that correct?
f(x) = 100x + 6


TO TELL IN ONE WORD 'CORRECT'. BUT TO EXPLAIN ...

WELL YOU CAN SAY SO,IF THAT IS THE PRACTICE THERE. NORMALLY IT IS CALLED SLOPE.
WHY I DO NOT ADVOCATE CALLING 'CONSTANT OF PROPORTIONALITY' IS THAT ,
SOME STUDENTS MAY MISINTERPRET IT AS TO CONCLUDE THAT Y [OR F(X) AS YOU PUT
IT] IS PROPORTIONAL TO X AND VARIES AS X WHICH IS NOT CORRECT.
ACTUALLY Y-6 IS PROPORTIONAL TO X AND VARIES AS X.
IF Y IS TO BE REALLY PROPORTIONAL TO X THEN THE EQN. HAS TO BE
Y=100X.......ONLY


In the following function, y varies as what power of x? y = 5 / x³ Thank you for all the help you have given me.
SINCE Y = 5/X^3 = CONSTANT/X^3…..WE SAY
Y VARIES INVERSELY AS X^3…THAT IS TO THE THIRD POWER OF X ……………….OR…………………
Y VARIES DIRECTLY AS X^(-3) …THAT IS TO THE –3 POWER OF X .