urgent--- test on tuesday --- chapter 6 inequality page 111

[tigerwoods]

Hi

Can you give me the algebraic breakdown of

for x2- x < 0
the solution says 0<x<1

but breakdown according to me is

x(x-1) <0

1) x <0

2) x-1 <0
x <1

then how did you 0<x<1 ......i understand that this is the right solution but my question is how did you derive it ?

but the solution says 0<x<1

am i loosing it......wats going on............

another doubt about page

page 115 it says a^2 -1 <0

why can we split it with

(a+1) (a-1) <0

hence

a<-1 or a <1 .... restrictive .... a <-1...but here the solution given in the guide which is corrrect is |a|<1..hence -1<a<1..... i get it....but I wanted to see why the above reasoning cannot work........

[StaceyKoprince]

Obviously, we are long past your test date - I'm sorry that we weren't able to get to this before your test date, but I did want to swing back around and answer it in case others have the same question in future. (Also FYI for future: it's really unlikely that you'll get an answer in less than 2 days; there's just too much forum traffic in general.)

I'll assume that x2 = x^2

You were right for part of the way:
x^2 - x < 0
x(x-1) < 0

So:
EITHER x < 0
OR x-1 < 0

These can't always be true at the same time for any quantity, because they're saying that (negative quantity #1) * (negative quantity #2) is negative. Can you multiply a negative by a negative and get a negative? (x) * (x-1) = negative? No. If both of those quantities are negative, then the product has to be positive. So, first, you need to insert "either / or" into your solutions, above.

So these two quantities multiply to some negative number. What qualities do you need to have in order to have two quantities multiply to a negative? One needs to be positive and one needs to be negative.

x < 0 is definitely negative, always, no matter what. So that has to be our negative quantity.
x-1 < 0 simplifies to x < 1. So x could be either positive or negative just according to this inequality. We need a positive quantity (to go with our x<0 negative quantity), so x has to be positive in this scenario. Positive is >0 by definition, so x is not only <1 but also >0. Therefore, 0 < x < 1.

Be really careful when you say >0 or <0 in a problem. Chances are, it's really trying to tell you something about positive and negative.

See if you can use that reasoning to figure out the answer to your second question.