Square root of variables

[TimH134]

Hi - I've finished a CAT exam, and missed a question on inequalities.

If -1 < x < 0, which must be true:

1) x^3 < x^2
2 x^5 < 1-x
3) x^4 < x^2

A) 1 only
B) 1 and 2 only
C) 2 and 3 only
D) 1 and 3 only
E) 1, 2, and 3

I solved for 1 and 2 correctly.

For #3, I took the square root of both sides: x^2 < x, and I set x = -1/2. This gave the wrong answer.

However, in some practice questions, I've come across statements asking whether x < 0, with a statement such as x^2 < 9. Here I took the square root, and came back with x < +/- 3. Why can I not do the same when it's variables on both sides of the inequality?

[Sage Pearce-Higgins]

You have to be careful when taking the square root of an inequality. It's simply a wrong algebraic move to see x^4 < x^2 and deduce x^2 < x. Try out a few examples to see that, for some numbers, the first will be true, but not the second. Your example of x = -1/2 is a good one, for this number x^2 is 1/4, and x^4 is 1/16, so x^4<x^2. It's the same for all numbers in the range -1<x<0.

There are patterns for exponents in 4 ranges: smaller than -1, between -1 and 0, between 0 and 1, and bigger than 1. The numbers -1, 0, and 1 are special cases. Pick some examples to see what these patterns are. And take a look at PS 67 from OG 2017.

[TimH134]

Thank you for your reply.

[Sage Pearce-Higgins]

No problem.