GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Dec 2019, 03:47 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x^2 − 2 < 0, which of the following specifies all the possible

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 59727
If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

6
36 00:00

Difficulty:   5% (low)

Question Stats: 77% (00:54) correct 23% (01:08) wrong based on 1439 sessions

### HideShow timer Statistics

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

Kudos for a correct solution.

_________________
Target Test Prep Representative G
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2809
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

10
14
Bunuel wrote:
If x^2 − 2 < 0, which of the following specifies all the possible values of x?

(A) 0 < x < 2
(B) 0 < x <
(C) $$-\sqrt{2} < x <\sqrt{2}$$
(D) −2 < x < 0
(E) −2 < x < 2

Kudos for a correct solution.

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

_________________

# Jeffrey Miller

Jeff@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager  P
Joined: 01 Mar 2015
Posts: 68
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

4
4
Bunuel wrote:
If x^2 − 2 < 0, which of the following specifies all the possible values of x?

(A) 0 < x < 2
(B) 0 < x <
(C) $$-\sqrt{2} < x <\sqrt{2}$$
(D) −2 < x < 0
(E) −2 < x < 2

x^2 − 2 < 0
=> x^2 < 2
=> mode of x < $$\sqrt{2}$$

therefore $$-\sqrt{2} < x <\sqrt{2}$$

##### General Discussion
Board of Directors P
Joined: 17 Jul 2014
Posts: 2491
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

1
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
Verbal Forum Moderator V
Status: Greatness begins beyond your comfort zone
Joined: 08 Dec 2013
Posts: 2435
Location: India
Concentration: General Management, Strategy
Schools: Kelley '20, ISB '19
GPA: 3.2
WE: Information Technology (Consulting)
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

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

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

_________________
When everything seems to be going against you, remember that the airplane takes off against the wind, not with it. - Henry Ford
The Moment You Think About Giving Up, Think Of The Reason Why You Held On So Long
Intern  Joined: 02 Oct 2015
Posts: 11
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

3
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
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4155
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

2
1
Bunuel wrote:
If x^2 − 2 < 0, which of the following specifies all the possible values of x?

(A) 0 < x < 2
(B) 0 < x <
(C) $$-\sqrt{2} < x <\sqrt{2}$$
(D) −2 < x < 0
(E) −2 < x < 2

If anyone is interested, we have a free video that explains how to solve quadratic inequalities such as this - http://www.gmatprepnow.com/module/gmat- ... /video/986

Cheers,
Brent
_________________
Manager  Joined: 13 Apr 2015
Posts: 73
Concentration: General Management, Strategy
GMAT 1: 620 Q47 V28 GPA: 3.25
WE: Project Management (Energy and Utilities)
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

mvictor wrote:
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

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..  Math Expert V
Joined: 02 Sep 2009
Posts: 59727
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

3
2
goldfinchmonster wrote:
mvictor wrote:
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

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..  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.
_________________
Senior Manager  G
Joined: 24 Nov 2015
Posts: 480
Location: United States (LA)
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

$$x^2$$ - 2 < 0
$$x^2$$< 2
As we know roots can be positive as well as as negative
Solution for $$x^2$$< 2
- $$\sqrt{2}$$ < x <$$\sqrt{2}$$
Intern  B
Joined: 22 Jun 2016
Posts: 3
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

\sqrt{}
Bunuel wrote:
goldfinchmonster wrote:
mvictor wrote:
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

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..  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.

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?
Math Expert V
Joined: 02 Sep 2009
Posts: 59727
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

2
3
YanisBoubenider wrote:
\sqrt{}
Bunuel wrote:
goldfinchmonster wrote:

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..  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.

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?

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 $$\sqrt{x^2}=\sqrt{25}=5=|-5|=|x|$$
_________________
Manager  S
Joined: 24 Sep 2011
Posts: 94
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

1
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
_________________
1. Well Begun is Half done
2. He who asks a question is a fool for five minutes; he who does not ask a question remains a fool forever.
3. The night is darkest just before the dawn
Math Expert V
Joined: 02 Sep 2009
Posts: 59727
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

1
Rocket7 wrote:
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

If you take the square root from x^2 < 2 you get $$|x| < \sqrt{2}$$, which in turn gives $$-\sqrt{2} < x <\sqrt{2}$$.
_________________
Non-Human User Joined: 09 Sep 2013
Posts: 13738
Re: If x^2 − 2 < 0, which of the following specifies all the possible  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: If x^2 − 2 < 0, which of the following specifies all the possible   [#permalink] 19 Feb 2019, 18:45
Display posts from previous: Sort by

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