Is x 5 (2) x^2 + x

Question

Is x<5 ?

(1) x^2 > 5

(2) x^2 + x < 5

harikrish · Answer

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.

Answer is Option B.

ydmuley · Answer

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 Hence, Answer is B

ScottTargetTestPrep · Answer

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.

Answer: B

MathRevolution · Answer

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.

Therefore, the answer is B.

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.

pranavpal · Answer

can anyone please solve the second statement in detail ?

ganand · Answer

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 eq 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.

krish76 · Answer

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

Probus · Answer

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

lstsch · Answer

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

Thank you in advance!

chetan2u · Answer

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

ScottTargetTestPrep · Answer

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.

Shef08 · Answer

Hi  Bunuel   VeritasKarishma ,

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

KarishmaB · Answer

(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.

Debo1988 · Answer

Bunuel VeritasKarishma

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?

GMATinsight · Answer

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

KarishmaB · Answer

Note that not every value in the range -3 < x < 2 will satisfy the original inequality x^2 + x < 5 (e.g. x = 1.9, -2.8 etc don't) 
though all values which will satisfy this inequality will fall in this range. 
Basically, by increasing 5 to 6, you have widened the actual range of values. 
So if x is less than 2 for x^2 + x < 6, it will certainly be less than 2 for x^2 + x < 5 too.

Also, it is not mandatory that the same approach needs to be followed in every question.

Yes, x^2 + x < 5 reminds me of a quadratic too but I also have my eye on x < 5 in the question stem. So I see that the only extra term stmtn 2 has is x^2 which I know cannot be negative. So even when I add 0 or a positive number to 5, the total is still less than 5. Then x must be less than 5. That's it!

aussieMBA · Answer

Can someone please explain the "Since x^2 is nonnegative, we have 5 - x^2 <= 5". I can only solve (2) by substituting numbers.  ScottTargetTestPrep

ScottTargetTestPrep · Answer

^2 is non-negative for any value of x, meaning it is either 0 or positive. Thus, when we write the expression 5 - x^2, we are subtracting something that is either 0 or positive from 5. If we subtracted 0, the result is 5 and if we subtracted a positive number, the result is less than 5. That's why 5 - x^2 ≤ 5.

It is also possible to obtain the same result by manipulating inequalities:

x^2 ≥ 0
-x^2 ≤ 0
5 - x^2 ≤ 5

BrentGMATPrepNow · Answer

Target question :    Is x<5 ?

Statement 1 : x² > 5  
Let's TEST some values.
There are several values of x that satisfy statement 1. Here are two: 
 Case a : x = -10, in which case the answer to the target question is  YES, x is less than 5 
 Case b : x = 10, in which case the answer to the target question is  NO, x is not less than 5 
Since we can’t answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : x² + x < 5 
Subtract x² from both sides of the inequality to get:   x < 5 - x²  
Since x² is always greater than or equal to 0, the GREATEST possible value of 5 - x² is 5, and this occurs when x = 0
So, when x is 0, the expression 5 - x² = 5
For all other values of x, we know that 5 - x² < 5
In other words,  5 - x² ≤ 5 
We can add this information to our existing inequality to get:   x < 5 - x²  ≤ 5 
At this point, we can clearly see that   x < 5  
The answer to the target question is  YES, x is less than 5 , which means statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

Is x<5 ? (1) x^2 > 5 (2) x^2 + x < 5

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