Reciprocals of inequalities

[cesar.rodriguez.blanco]

I do not understand the "Reciprocals of inequalities", page 119 Guide 3, 3rd Ed, Equations, Inequalities and VIC.

For example, the guide says:
"Taking reciprocals of inequalities is similar to multiplying/ dividing by negatives numbers".

But it is not true because we have to consider always the sign of the values. The guide explains the case x<y, but not x>y

Furthermore, I do not understand the next example, when the guide says:
if a>b, then 1/a < 1/b, unless a and b have different signs......

Thanks

[Ben Ku]

I think this principle is more clear if you think about the PRINCIPLES involved instead of memorizing the properties.

Given that x < y. Suppose both x and y are positive. To compare the reciprocals, we are essentially dividing both sides by x, and by y.
x < y
1 < y/x (Divide by x)
1/y < 1/x (Divide by y)

Now Given that x < y. Suppose x is positive and y is negative. We'll do the same:
x < y
1 < y/x (Divide by x)
1/y > 1/x (Divide by y, which is negative)

I think if you think of these properties in this way, it's clear what happens to the inequality. Let me know if you have additional questions.

[cesar.rodriguez.blanco]

One question:

I do not understand the second explanation because it is impossible that a negative number be bigger than a positive number, so could you reformulate your explanation?

[kevinmarmstrong]

Suppose x < y (1) and xy > 0.

Dividing both sides of (1) by xy, we get 1/y < 1/x
(Remember when you divide both sides of an inequality by a positive number, the inequality sign remains unchanged)


Suppose instead that xy < 0 (i.e. x and y have opposite signs)
Dividing both sides of (1) by xy, we get 1/y > 1/x
(Remember when you divide both sides of an inequality by a negative number, the inequality sign changes)

[Ben Ku]

Now Given that x < y. Suppose x is positive and y is negative. We'll do the same:
x < y
1 < y/x (Divide by x)
1/y > 1/x (Divide by y, which is negative)

One question:

I do not understand the second explanation because it is impossible that a negative number be bigger than a positive number, so could you reformulate your explanation?

Sorry Cesar. I meant to say:

Suppose x is negative and y is positive.

In that case it makes sense that x < y.

Hope that helps!