If x^2 2

Question

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

(A) 0 < x < 2

(B)  0 < x < \sqrt{2}

(C)  -\sqrt{2} < x <\sqrt{2}

(D) −2 < x < 0

(E) −2 < x < 2

JeffTargetTestPrep · Accepted Answer

Solution:

We must determine all of the values of x based on the inequality x^2 – 2 < 0. To determine all the possible values of x, let’s simplify the inequality.

x^2 < 2

√x^2 < √2

|x| < √2

Note that when we take the square root of x^2, the value of x itself can be either negative or positive, and thus we express this as |x|, the absolute value of x.

Remember also that when we are solving an absolute value equation or inequality, we consider two cases: when the quantity inside the absolute value is positive and then when the quantity inside the absolute value is negative.

Let's now solve for when x is positive and then for when x is negative.

When x is positive

x < √2

When x is negative

-x < √2

x > -√2

Combining these two inequalities, we have:

-√2 < x < √2

Answer: C

Bunuel · Answer

When the GMAT provides the square root sign for an even root, such as \sqrt{x} or \sqrt {x} , then the only accepted answer is the positive root. That is, \sqrt{16}=4 , NOT +4 or -4. Even roots have only a positive value on the GMAT. In contrast, the equation x^2=16 has TWO solutions, +4 and -4. Odd roots have the same sign as the base of the root. For example, \sqrt {125} =5 and \sqrt {-64} =-4 . Now, about the problem itself: MUST KNOW: \sqrt{x^2}=|x| : The point here is that since square root function cannot give negative result then \sqrt{some \ expression}\geq{0} . So \sqrt{x^2}\geq{0} . But what does \sqrt{x^2} equal to? Let's consider following examples: If x=5 --> \sqrt{x^2}=\sqrt{25}=5=x=positive ; If x=-5 --> \sqrt{x^2}=\sqrt{25}=5=-x=positive . So we got that: \sqrt{x^2}=x , if x\geq{0} ; \sqrt{x^2}=-x , if x<0 . What function does exactly the same thing? The absolute value function: |x|=x , if x\geq{0} and |x|=-x , if x<0 . That is why \sqrt{x^2}=|x| . For example if x = -5, then \sqrt{x^2}=\sqrt{25}=5=|-5|=|x|

mvictor · Answer

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

we can see that x^2 is less than 2. it can be the case that sqrt(2) must be less than 2.
since the square of any negative integer is a positive number, we should take into consideration that x cannot be less than -sqrt(2), but not greater than sqrt(2)
the only answer here is C

v12345 · Answer

x^2 − 2 < 0 => x^2 < 2 => mode of x < \sqrt{2} therefore -\sqrt{2} < x <\sqrt{2} Answer option C

Skywalker18 · Answer

x^ {2}  -2 < 0
=> x^ {2} <2

Therefore , -  \sqrt{(2)} < x < \sqrt{(2)}

Answer C

ramajha · Answer

y=x^2-2=(x-sqrt2)*(x+sqrt2) when x in in between these two values (-sqrt2,sqrt2) ,y will be negative.
So Option C

BrentGMATPrepNow · Answer

STRATEGY : Upon reading any GMAT Problem Solving question, we should always ask,   Can I use the answer choices to my advantage?   
In this case, we can easily test values that satisfy the given inequality. 
From here, I'll give myself 15-20 seconds to identify a faster approach..... 
We can also solve the question algebraically, but since I'm less likely to make mistakes testing values, I'll go that route

I can automatically see that  x = 0  is a convenient solution to the given inequality, x² − 2 < 0
So, let's plug   x = 0   into each answer choice to see if it's a solution.
(A) 0 <   0   < 2. Doesn't work. Eliminate A.
(B) 0 <   0   < √2. Doesn't work. Eliminate B
(C) -√2 <   0   <√2. Works. KEEP.
(D) −2 <   0   < 0. Doesn't work. Eliminate D
(E) −2 <   0   < 2. Works. KEEP.

After testing this convenient x- value, we are down to answer choices C and E.

TIP: By test day, all students should have the following approximations memorized:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2

Since √2 ≈ 1.4, we can see that answer choice C tells us that x CAN'T equal 1.5, and answer choice E tells us that x CAN equal 1.5.
So let's plug   x = 1.5   into the given inequality,  x² − 2 < 0 
When we do so, we get:   (1.5) ² − 2 < 0 , which simplifies to be:  2.25 - 2 < 0 , which is FALSE. 
Since   x = 1.5   is NOT a solution to the given inequality, we can eliminate answer choice E, because it says   x = 1.5   IS a solution.

By the process of elimination, the correct answer must be C

goldfinchmonster · Answer

Hi.. According to me X^2 < 2 can be written as Mod (X) < 2, which in tern yields the range of x as -2 < x < 2... Could you tell me where iam going wrong.. Thanks in advance..

Bunuel · Answer

If you take the square root from x^2 < 2 you get  |x| < \sqrt{2} , which in turn gives  -\sqrt{2} < x <\sqrt{2} .

Hope it's clear.

YanisBoubenider · Answer

\sqrt{}

Hi Bunuel

I'm struggling a little with this one.

Indeed I assumed that when GMAT gives the symbol of a square root, we should only consider the positive number of x. In other words if GMAT asks x^16 therefore X = 4 or x = -4 BUT if GMAT asks √16 therefore we should only consider 4 and NOT -4. 
Coming back to the question, I assumed that x^2 < 2 and therefore √x<√2 and therefore x must be strictly positive and be between 0 and √2

Even though I kind of understand your reasoning, where am I doing wrong here?

Rocket7 · Answer

hello

Can anyone explain why E is wrong? I understand why C is right and I understand the way modulus is functioning in this Q. so when we have x^2<2. cant we just say that x must be -2<x<2

Bunuel · Answer

If you take the square root from x^2 < 2 you get  |x| < \sqrt{2} , which in turn gives  -\sqrt{2} < x <\sqrt{2} .

keshavmishra · Answer

given x^2-2<0
break as a^2-b^2=(a-b)(a+b)

(x-Sq. root2)(x+sq. root 2)<0
Product of 2 can be negative, if both terms are of opposite sign.

x>-Sq.root2 & X<sq.root2
rance -sq.root2<x<sq.root2

Answer C

Posted from my mobile device

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=XbFYjhUkRB0

CrackverbalGMAT · Answer

Solution: x^2 − 2 < 0 => x^2 < 2 If x is positive then x < √2 If x is negative then -x < √2 => x > -√2 Combining them, -√2 < x < √2 (option c) Devmitra Sen GMAT SME

GMATinsight · Answer

Answer: Option C

Video solution by  GMATinsight

https://www.youtube.com/watch?v=L-nAt5Xum0I

bumpbot · Answer

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If x^2 2 < 0, which of the following specifies all the possible

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