Question: If x is not equal to 0, is –1 ≤ x ≤ 1 ?

(1) x^3 < x^2
(2) x^2 < x

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Answer: D
Solution and Explanation:
Approach Solution 1:
Given in the question that x is not equal to 0. It has been asked whether we can say that -1 < x < 1 from the given statements.
Given in statement 1, that \(x^3 < x^2\)
=> \(x^3 - x^2 < 0\)
=>\(x^2 (x-1) < 0\)
=> \(x^2(x-1) < 0\)
We get x < 1
X will be less than 1 but may not be greater than -1.
Hence this statement is not sufficient.
Given in statement 2,
\(x^2 < x\)
\(x^2 - x < 0\)
\(x(x-1) < 0\)
Hence x will lie between (0,1)
As we know that x will lie in the range (0,1) then we can also say that x lies between (-1,1)

Approach Solution 2:
Given in the question that x is not equal to 0. It has been asked whether we can say that -1 < x < 1 from the given statements.
Statement 1: x^3 < x^2

• x^3−x^2 < 0
• x^2(x−1) < 0

We know that x^2 is greater than 0.
Therefore, x−1<0
and x<1
However, we don't know whether x > -1 or not.
Hence, statement 1 is not sufficient, and we can eliminate answer options A and D.
Statement 2: x^2 < x
We know that x^2 is always positive
Thus, x is also positive because x2<x
This is only possible when x is between 0 and 1.
0<x<1
It means −1≤ x ≤1
Hence, statement 2 is sufficient, so the correct answer is option B.

Approach Solution 3:
Given in the question that x is not equal to 0. It has been asked whether we can say that -1 < x < 1 from the given statements.
If x has a negative value, then x3 will always be an inferior product to x2 For instance, if x equals -1/2, then x2 equals 1/2 and x3 equals -1/8. Once more, when x equals -2, x2 equals 4, and x3 equals -8. Not sufficient.
For value from 0 to 1, x^2 < x. e.g: x =1/4, x^2 = 1/16 or x =1/3, x^2=1/9. So, x is between 0 and 1. Sufficient.
The correct response is B.

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