alphabeta1234 wrote:

Hi,

Is there any general rule for multiplication or division of two inequalities?

If

a<x<b and c<y<d

then

a+c<x+y<b+d

a-d<x-y<b-c

Now is there any rule to find the relationships for xy or x/y?

Thank you in advance!

Multiplication - Yes but very constrained.

If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them.

x < a

y < b

xy < ab

Given x, y, a, b are all positive.

Otherwise

-2 < -1

10 < 30

Multiply: -20 < -30 (Not correct)

or

-2 < 7

-8 < 1

Multiply: 16 < 7 (Not correct)

For division, this may not hold.

e.g

2 < 10

4 < 40

Divide: 1/2 < 1/4

But if both sides of both the inequalities are positive and the signs of the inequality are opposite, then you can divide them

x < a

y > b

x/y < a/b (given all x, y, a, b are positive)

The final inequality takes the sign of the numerator. Take examples.

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