Question: If x^2 − 2 < 0, which of the following specifies all the possible values of x?

(A) 0 < x < 2
(B) 0 < x < √2
(C) −√2 < x < √2
(D) −2 < x < 0
(E) −2 < x < 2

Correct Answer: (C)
Solutions and Explanation
Approach Solution - 1:

On the basis of the inequality x^2 − 2 < 0, we must ascertain all possible values for x. Let's simplify the inequality to find all possible values for x.
x^2 < 2
√x^2 < √2
|x| < √2
Because the value of x itself can be either positive or negative when we take the square root of x^2, we express this as |x|, the absolute value of x.
Also keep in mind that there are two scenarios we must take into account when solving an absolute value equation or inequality. They are the positive and negative values of the quantity inside the absolute value.
Now that x is positive, let's solve for it, and after that, for x when it is negative.
For x as positive:
x < √2
For x as negative
-x < √2
=> x > -√2
By combining these two inequalities, we will get −√2 < x < √2.

Approach Solution - 2 :
We can immediately see that the given inequality x^2 − 2 < 0, has a convenient solution of x = 0.
So, let's try each option with x = 0 to see if it provides a solution.
When substituted into all the equations in the options, it is clear that the first, second and fourth options do not work and therefore they are eliminated.
We can see that the third option states that x cannot be equal to 1.5 and the fifth option states that x can be equal to 1.5. This is said because √2 ≈ 1.4.
In order to solve the given inequality x^2 − 2 < 0, let's insert x = 1.5.
When we do this, we obtain [(1.5)^2] - 2 < 0, which is false when simplified to be 2.25 - 2 < 0.
We can rule out the last option because it claims that x = 1.5 is a solution, despite the fact that x = 1.5 is not a solution to the given inequality.

Approach Solution - 3 :
It is obvious that x^2 is less than 2. It can be the scenario where √2 has to be less than 2.
Given that the square of any negative integer is a positive number, we should consider the fact that x cannot be either greater than √2 or less than -√2 .
Therefore, −√2<x<√2 is the only answer.
“If x^2 − 2 < 0, which of the following specifies all the possible” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

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