Please show that the absolute value of absolute value of x minus absolute value of y less than or equal to absolute value of x minus y.?

Answer 1

to prove x|-|y≤|x-y| have to look at 4 cases:

a) x>0, y>0

b) x<0, y<0

c) x>0, y<0

d) x<0, y>0

(to save typing it out over and over again, i have shortened absolutes to abs)

For: a) the values dont matter both sides will always have the same value.

b) let x=-a, y=-b. Because of the abs around x and y, the left will be a-b and the right b-a so the left and right will have the same number but opposite signs, so after taking the abs on both sides, they will always be equal to one another.

c) let x=a, y=-b. The left will be a-b and the right a+b. if b<a then both sides are positive and the statement holds. if a<b then left is negative and right positive, but because you're starting from above 0 when you take away b (on the left side) once you take the abs left will be less than the right. when a=b the left is just 0 and right is positive so statement still holds.

d) let x=-a, y=b. left is a-b again, but the right is-a-b so always negative. if a<b then a-b is negative but not as low as -a-b so left is less than right after taking abs. if b<a then a-b is positive, but moving closer to 0 and -a-b moves further away from 0 so left is less than right again after taking abs. If a=b then then the left is always 0 and after taking abs the right is 2a therefore the statement holds for all 4 cases. Q.E.D.