If –1 < x < 1 and x ≠ 0, which of the following inequalities must be true?
I. x^3 < x
II. x^2 < |x|
III. x^4 – x^5 > x^3 – x
Options:
A) I only  B) II only  C) III only  D) II and III only  E) I, II, and III
Answer is D. Let's look at the statements one by one   Stmt I. x^3 &lt; x   if 0<x<1 -1<x<0="" but="" if="" then="" x="" x^3="" x^3<="">x   So this statement is not always true   Stmt II. x^2 &lt; |x|   Because we know that x is a number less than one but not equal to zero then x^2 will always be less than |x|.   Why? think of positive fractions (and you can think in terms of positive fractions because the inequality is in regards to |x|). Lets set x </x<1>= 1/2, then x^2 = 1/4 and 1/4&lt;1/2   So Stmt II is always true   Stmt III. x^4 – x^5 &gt; x^3 – x^2   This one may seem tricky but lets break it down. x^4 – x^5 &gt; x^3 – x^2 = x^4(1-x)&gt;x^2(x-1).   Because lets concentrate on (1-x) and (x-1). We are given that -1<x<1 (1-x)="" means="" that="" which="">0 and (x-1)&lt;0. x^4 will always be positive and x^2 will always be positive so without doing any math we are looking at positive &gt; negative... which is always true.   So Stmt III is always true   I only</x<1>
<result>A</result>
If –1 < x < 1 and x ≠ 0, which of the following inequalities must be true?
I. x^3 < x
II. x^2 < |x|
III. x^4 – x^5 > x^3 – x