QC & DI, chapter 3 Algebra, page 43 and 45-46

[danc]

In the Relationship Only section on page 43, for the example given, if it were an inequality rather than an equation, can you still use cross-multiplication?

On page 45-46, once we FOIL quantity A and subtract 5pq from each quantity, why don't we subtract a p^2 and a q^2 from each quantity? That would leave us with p^2+q^2 for quantity A and 0 for quantity B, and because pq cannot equal zero we know neither variable is 0, so A must be greater. Is there a reason not to do that? Does it have to be the entire term in common to be subtracted?

[tommywallach]

To the first question, I don't understand. They are doing the hidden inequality method there. I don't see any equations. Could you write out here what equation you're talking about?

To the second question, yes, you could absolutely subtract p^2 and q^2 from both columns. You shouldn't technically need to at that point, but it's totally correct to do so.

-t

[danc]

The equation at the top of page 43 is (x+5)/5=(y+6)/6, and they cross multiply before subtracting 30 from each side to simplify the equation. It's not the hidden inequality; it is the given information. Quantity A is 6x and quantity B is 5y. My question is if we get a question like this except instead of an = it is an inequality, does cross multiplication still work?

[tommywallach]

Hey Dan,

Aha! Now I understand! With that particular equation, the answer is YES. That's because when we cross-multiply, we are actually multiplying both sides by both denominators (of the two fractions in question). That's what cross-multiplication is.

In this case, the denominators are numbers, so it's totally legal to cross multiply even in an inequality. We can't cross-multiply in any fraction where there are denominators with unknown signs (positive/negative), because that might flip the inequality symbol.

Hope that helps!

-t

[danc]

Cool. Thanks. Does that mean if we know that one denominator is negative, then we can cross multiply but must flip the inequality? And if we know both denominators are negative, we cross multiply and keep the inequality the same?

[tommywallach]

Yes to both of your questions! That said, I would caution you to be very careful when applying this logic to the "giant imaginary inequality method" of doing QC. In that case, if the sign "flips" (i.e. if you multiply both columns by a number you KNOW to be negative), whichever column now looks bigger is ACTUALLY the smaller column. This can be really confusing, of course, so feel free to test some numbers to be sure of what's happened.

-t