<p>I got this answer right, but I feel I spent more time than I should have, and could have just as easily got it wrong with one mistake. </p>
<p>Each of the following inequalities is true for some values of x EXCEPT</p>
<p>A. x < x^2 < x^3
B. x < x^3 < x^2
C. x^2 < x^3 < x
D. x^3 < x < x^2
E. x^3 < x^2 < x</p>
<p>Is it B? i did it in less than 30 seconds.</p>
<p>No its C :( Its a hard question too.</p>
<p>A works for x > 1
B works for -1 < x < 0
C works for none
D works for x < -1
E works for 0 < x < 1</p>
<p>How did you figure all that out and how quickly? I mean, do you just "know" these facts and constructions? Or did you plug in numbers? Or what?</p>
<p>It definitely helps to know the properties of f(x)=x^(p) for the following values of p:
1/3, 1/2, 1, 2, 3.
Graph them and compare their "behavior" for
x<-1
-1=<x<0 0="<x<1" and="" 1="<x." (domain="" of="" x^(1="" 2)="" is="" x="">=0).
Make a table with a column for each interval and a row for each p.
Pug some value for x from each interval into f(x)=x^p for each p and fill out the table.
See how these functions are interrelated?
Now you see the "Fast way to approach this problem".
And not only this one.
Check out 550/14, 585/13, and 745/8 on the Blue Book.</x<0></p>
<p>by cases, try x > 1, 0<x<1, -1<x<0, and x<-1 for each, these kinds of problems come up over and over, you don't have to pick specific numbers, just think in terms of magnitude and sign and that should point you in the right direction.</p>
<p>
Absolutely.
You do that once at home, as I described, and memorize the picture - which graph is above/below which for the rest of your life.</p>
<p>I think if you are going to look at specific cases, because you don't know the pattern the day of the test, you should always look and make sure you are checking negative numbers, 0, and fractions, along with the usual positive numbers.</p>
<p>On the contrary, I think the easiest way for most students to do these kinds of questions is to plug in different x's. It's much easier not to make a mistake that way. If you've found x's that work for all of the inequalities except one, you know you're right. Knowing the general shapes of curves is a good idea, but it's easy to forget special cases like negative fractions, positive fractions, and 0.</p>
<p>Here's how I'd do it:</p>
<p>You know four of them are correct, and you know there's four different "types" of numbers that x can be. It can be a negative greater than one, negative less than one, positive less than one, or positive greater than one.</p>
<p>So... just use -2,-1/2, 1/2, 2 and find the x,x^2,x^3 values for all four, and rank them using inequalities. Each one matches one letter, leaving one answer left over.</p>
<p>Hope this helps.</p>