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Director  B
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Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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Question Stats: 72% (01:44) correct 28% (01:54) wrong based on 1479 sessions

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Is x<5 ?

(1) x^2 > 5

(2) x^2 + x < 5
Target Test Prep Representative V
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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13
8
AbdurRakib wrote:
Is x<5 ?

(1) x^2 > 5

(2) x^2 + x < 5

We need to determine whether x < 5.

Statement One Alone:

x^2 > 5

If x = 3, then x is less than 5. However, if x = 6, then x is not less than 5. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x^2 + x < 5

Thus, we have x < 5 - x^2. Since x^2 is nonnegative, we have 5 - x^2 ≤ 5. Since x < 5 - x^2 and 5 - x^2 ≤ 5, we have x < 5.

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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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24
9
AbdurRakib wrote:
Is x<5 ?

(1) $$x^2$$>5

(2) $$x^2$$+x<5

(1) $$x^2>5$$

If $$x = 6$$ or$$-6$$. $$x^2$$ will be greater than 5 in both cases.

However 6 is greater than 5 and -6 is less than 5.

(1) has multiple values. Hence I is Not Sufficient.

(2) $$x^2+x<5$$

$$x(x+1) < 5$$

$$x<5$$ or

$$x+1<5 = x < 4$$

(2) has x less than 5 or 4. Therefore x is less than 5. II is Sufficient. Answer (B)...
##### General Discussion
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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2
AbdurRakib wrote:
Is x<5 ?

(1) $$x^2$$>5

(2) $$x^2$$+x<5

Solution:

Statement 1: x can be 3,4,5 or 10. Insufficient.

Statement 2: The greatest positive value that satisfies the equation is 1. Therefore, x<5 . Sufficient.

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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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4
1
Is x<5 ?

$$(1) x^2 > 5$$

This means x can have Positive and Negative values , lets check

$$x = 10 = x^2 = 100$$

$$x = - 10 = x^2 = 100$$

As we are getting answer as YES & NO

Eq. (1) =====> is NOT SUFFICIENT

$$(2) x^2 + x < 5$$

$$x^2 + x < 5$$

$$x(x + 1) < 5$$

$$x < 5$$ or

$$x < 4$$

As both these values are < 5

(2) =====> is SUFFICIENT

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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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2
1
AbdurRakib wrote:
Is x<5 ?

(1) x^2 > 5

(2) x^2 + x < 5

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variables and 0 equations, D is most likely to be the answer and so we should consider each of conditions first.

Condition 1)
$$x = 10$$ : Yes
$$x = -10$$ : No
This is not sufficient.

Condition 2)
$$x^2 + x < 5$$
$$x < 5 - x^2 < 5$$
This is sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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ydmuley wrote:
Is x<5 ?

$$(2) x^2 + x < 5$$

$$x^2 + x < 5$$

$$x(x + 1) < 5$$

$$x < 5$$ or

$$x < 4$$

As both these values are < 5

(2) =====> is SUFFICIENT

can anyone please solve the second statement in detail ?
Manager  D
Joined: 17 May 2015
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Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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3
pranavpal ,

St(2): $$x^{2} + x < 5$$

There are a couple of ways, we can solve the second statement.

Method1: Use Quadratic equation root formula:

For a given Quadratic equation $$ax^{2} + bx + c = 0$$ (where $$a \neq 0$$), roots can be obtained using following frmula:

$$\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

Lets first solve the equality case of st(2) i.e. : $$x^{2} + x -5 = 0$$ . Here we have, a = 1, b = 1, c = -5. Put all the values in the above formula, we get:

Roots = $$\frac{-1 - \sqrt{21}}{2}, ~~~~ \frac{-1 + \sqrt{21}}{2}$$ => approximately roots are -3 and +2.

In order to solve the inequality, put the root's value on the number line and check whether the inequality gets satisfied or not in each section.

----------------Not satisfy ----------(-3) --------Satisfy--------(2)--------Not satisfy------------

It is very clear that the st(2) will satisfy only when -3 < x < 2. => x is definitely less than 5. Hence, Sufficient.

I hope this helps.

Thanks.
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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9
Although most of the folks got it right, I want to mention something which most of the folks got wrong..

x(x+1)<5

DOES NOT MEAN - x < 5 or (x+1) < 5.

and you easily check it by substitution

See bunuel's post - https://gmatclub.com/forum/inequalities ... 06653.html
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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1
pranavpal wrote:
ydmuley wrote:
Is x<5 ?

$$(2) x^2 + x < 5$$

$$x^2 + x < 5$$

$$x(x + 1) < 5$$

$$x < 5$$ or

$$x < 4$$

As both these values are < 5

(2) =====> is SUFFICIENT

can anyone please solve the second statement in detail ?

Hi,

The second statemnt if we read it along we can understand if this would be sufficient.

We are told that square of a number added to the number itself is less than 5. that means at least x< 5 and also that at least $$x^2< 5$$

Probus
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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Hi all,

I understand that plugging in values is a very good approach for this question. Still, I am wondering how to solve Statement 1 algebraically.

My way:

(1) x^2 > 5
|x| > √5 (correct?)
x > √5 or x < -√5 --> Therefore x can be bigger than 5, insufficient

Thanks a lot!
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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Can someone please show the exact steps from: "Since x < 5 - x^2 and 5 - x^2 ≤ 5, we have x < 5."
Why is x ≤ 5 not possible? Or how do I know which sign I should take when adding inequalities?

ScottTargetTestPrep Bunuel chetan2u

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Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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1
lstsch wrote:
Can someone please show the exact steps from: "Since x < 5 - x^2 and 5 - x^2 ≤ 5, we have x < 5."
Why is x ≤ 5 not possible? Or how do I know which sign I should take when adding inequalities?

ScottTargetTestPrep Bunuel chetan2u

$$x < 5 - x^2$$ and $$5 - x^2 ≤ 5$$
When we combine the two, we get $$x< 5 - x^2 ≤ 5$$... This, x< 5 - x^2 ≤ 5 gives us x<5
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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2
1
First of all, we are NOT adding the two inequalities to arrive to the conclusion that x < 5. Let’s be clear on that. It’s a transitive property in inequalities.

For example, if a < b and b < c, then a < c. Another example is: if a ≤ b and b ≤ c, then a ≤ c. Of course, here we have if a < b and b ≤ c, then a < c. You can see that the premise actually can be combined into a double inequality, that is, for short, we can say: a < b < c implies that a < c; a ≤ b ≤ c implies that a ≤ c; and last but not least, a < b ≤ c implies a < c.

So your question is, if a double inequality have both < and ≤ signs, why we take the < sign, instead of the ≤ sign?

The reason is we always take the sign of the < (“strictly less than”) sign when a double inequality have both. That is because in a < b ≤ c, it says b is no more than c, so if a is strictly less than b, it will be also strictly less than c, hence the conclusion inequality a < c. The other way around is also true.

That is, a ≤ b < c also implies a < c. Here, it means a is no more than b, but if b is strictly less than c, so does a.
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Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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I need ur expert opinion in this..

I got the answer correct but need to know whether my process is correct or no..

S2: x^2+x<5
X^2+x-5<0
(X+1)(x-5)<0

X<-1 or x<5

Posted from my mobile device
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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Shef08 wrote:

I need ur expert opinion in this..

I got the answer correct but need to know whether my process is correct or no..

S2: x^2+x<5
X^2+x-5<0
(X+1)(x-5)<0

X<-1 or x<5

Posted from my mobile device

(x + 1)*(x - 5) is not the same as (x^2 + x - 5). It is same as (x^2 - 4x -5).

(x^2 + x - 5) does not have integer roots but we can guess the roots using the formula as done in this comment above: https://gmatclub.com/forum/is-x-5-1-x-2 ... l#p2042359

If the roots are approximately -3 and 2, then (x^2 + x - 5) = (x + 3)(x - 2) < 0
which gives us -3 < x < 2
So in any case, x is less than 5.
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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What if we plug in -10 to the second inequation? In that case, the result will be more than 5 and B will be wrong as well, IMO. Could someone clarify this?
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GMAT 1: 630 Q43 V34 Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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Hi! I have read all the explanations above and I stil don’t get the point of the 2nd statement explanation. Can anyone explain it pls?

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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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Can we use the following approach for statement 2?

statement 2 : x^2 + x < 5
As soon as i see this statement, my instinct tells me to break it in the form (x-a)(x-b)<0, which will give me a range for x.
But we can't do that easily because x^2 + x -5 doesn't yield integral roots.
Instead can we proceed as below ? :
x^2 + x < 5
=> x^2 + x < 6 (combining the 2 inequalities : x^2 + x < 5 and 5 < 6 )
=> (x-2)(x+3) < 0
=> -3 < x < 2 (Sufficient)

Do you see any risk in such an approach?
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Re: Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5  [#permalink]

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Debo1988 wrote:

Can we use the following approach for statement 2?

statement 2 : x^2 + x < 5
As soon as i see this statement, my instinct tells me to break it in the form (x-a)(x-b)<0, which will give me a range for x.
But we can't do that easily because x^2 + x -5 doesn't yield integral roots.
Instead can we proceed as below ? :
x^2 + x < 5
=> x^2 + x < 6 (combining the 2 inequalities : x^2 + x < 5 and 5 < 6 )
=> (x-2)(x+3) < 0
=> -3 < x < 2 (Sufficient)

Do you see any risk in such an approach?

Debo1988
Absolutely no risk in this approach. It's algebraically Sound _________________
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