Question

If x and y are real numbers such that x > 1 and y < −1,

then which of the following inequalities must be true?
A. x/y> 1
B. |s|^2 > |y|
C. x/3− 5 > y/3 − 5
D. x^2 + 1 > y^2 + 1
E. x^(−2) > y^(−2)

Answer 1

`x > 1\ \wedge\ y < -1\\\\A.\ (x)/(y) > 1-FALSE;example:x=2;\ y=-2\ then\\L=(2)/(-2)=-1;R=1;\ L < R\\\\B.\ |x|^2 > |y|-FALSE;example:x=2;\ y=-6\ then\\L=|2|^2=2^2=4;R=|-6|=6;\ L < R\\\\C.\ (x)/(3)-5 > (y)/(3)-5-TRUE;(x)/(3)-5 > (y)/(3)-5\to(x)/(3) > (y)/(3)\to x > y`


`D.\ x^2+1 > y^2+1-FALSE;example:x=2;\ y=-3\ then\\L=2^2+1=4+1=5;R=(-3)^2+1=9+1=10;L < R\\\\E.\ x^(-2) > y^(-2)-FALSE;example:x=3;\ y=-2\ then\\L=3^(-2)=(1)/(9);R=(-2)^(-2)=(1)/(4);\ L < R`