<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>How do I approach this type of question?</p>
<p>Pretty much just plug in values for x. If even one value of x makes the inequality true, it’s not the answer. </p>
<p>a) x < x^2 < x^3
You shouldn’t even need to plug anything in here. This is true for any x > 1</p>
<p>b) x < x^3 < x^2
Now’s where it gets trickier. You’re going to have to plug in fractions, negative numbers, etc. in hopes of getting a match. Let’s let x = -1/2. Then -1/2 < -1/8 < 1/4</p>
<p>c) x^2 < x^3 < x
There doesn’t seem to be anything here, so it must be the answer.</p>
<p>d) x^3 < x < x^2
x = -2. -8 < -2 < 4</p>
<p>e) x^3 < x^2 < x
x = 1/2. 1/8 < 1/4 < 1/2</p>
<p>Remember, even ONE value of x that makes the inequality true will eliminate that answer choice. How do you pick values for x? It’s just about practice. Cubing or squaring a positive integer > 1 makes it larger, but cubing or squaring a fraction can also make numbers smaller too (e.g. x = 1/2). Negative signs throw things off too. Practice, practice, practice.</p>
<p>Usually all you have to know is that squaring and cubing fractions make them smaller, and cubing negative numbers keeps them negative.</p>
<p>The main thing to do for these (whenever you see x^2 x^3 and this kind of stuff) is pick 4 sets of numbers: Positive > 1, Positive < 1, Negative > 1, Negative <1.
The ones less than one should be a fraction.
So you may pick x= 2, -2, 1/2, and -1/2.
Then just plug the numbers in.
Or you can just use logic if you can just see it in your head without using the exact numbers.</p>
<p>Yeah by rule of hand, just try 2, 0.5, -0.5, and -2.</p>
<p>These questions come up often, so after enough practice, you’ll be a pro at identifying this in your head.</p>