If x and y; and x>y then -1/x > -1/y? Prove.?

Question

If x and y are nonzero real #s and x>y then (-1/x) > (-1/y). Show where this is false and add a hypothesis on y that makes the statement true.

I know that if 0

Answer 1

Given: x > y and x ≠ 0, y ≠ 0.
pf.

Case 1: Let x and y be positive Real numbers.
x > y
-x < -y [Multiply both sides by -1.]
-1 < -y/x [Divide both sides by x.]
-1/y < -1/x. [Divide both sides by y.]
-1/x > -1/y.

Case 2: Suppose x > 0 and y < 0.
x > y
-x < -y
-1 < -y/x
-1/y > -1/x
-1/x < -1/y.
[Case 2 fails to yield the desired conclusion.]

Case 3: Suppose x < 0 and y < 0.
x > y
-x < -y
-1 > -y/x
-1/y < -1/x
-1/x > -1/y.

The result holds provided y > 0, whenever, x> 0.

Answer 2

Why not try this for all 4 cases.
Case 1: x > 0 and y > 0
Given x > y
Divide both sides by x
1 > y/x
Divide both sides by -y
(remember to flip the sign because -y < 0)
-1/y < -1/x or -1/x > -1/y
So it is true for x and y > 0

Case 2: x > 0, y < 0
Given x > y
Divide both sides by x
1 > y/x
Divide both sides by -y
(you do not need to flip the sign because -y > 0)
-1/y > -1/x or -1/x < -1/y
So the statement -1/x > -1/y is false.

Do this for
Case 3: x <0, y > 0 and Case 4: x < 0, y < 0
and you should have your proof for where this is false and where this is true.

Answer 3

If x and y; and x>y then 1< x/y, since x/y is an improper fraction.
Thus 1/x < 1/y [divide both sides by x]
Thus -1/x > -1/y [ multiply by -1 changes < to >.

Answer 4

What if x is positive and y is negative? Then -1/y is a positive number and -1/x is a negative number and therefore -1/y > -1/x and not the reverse. I think y must be positive.

Answer 5

Given
x > y
So >> 1/x < 1/y
Then >>>>> - 1/x > - 1/y

Answer 6

you have
x>y
you can't say that
1/y<1/x
b/c it means dividing the inequality by x and then by y, and if you divide by a negative number, the sign of inequality flips.
So it is only true if x and y have same sign.

If they do, then you multiply by -1 and get the result.

Answer 7

uhm, i imagine it really is in uncomplicated words as uncomplicated of basically declaring this: if x

Answer 8

Just about any counterexample of x and y where y<0