If x is an integer and 2x^2 + 9

Question

If  x  is an integer and  2x^2+9 < 9x , how many positive factors does  x  have?

A. 1
B. 2
C. 3
D. 4
E. 5

Bunuel · Accepted Answer

Official Solution:

If  x  is an integer and  2x^2+9 < 9x , how many positive factors does  x  have?

A. 1
B. 2
C. 3
D. 4
E. 5

First, rearrange the inequality:  2x^2 - 9x + 9 < 0 . Now, factor the quadratic:  (2x - 3)(x - 3) < 0 . The roots are  \frac{3}{2}  and 3. Since the inequality sign is "  <  ", the solution lies between the roots:  1.5 < x < 3 . The only integer in this range is 2, so  x=2 , which has two positive factors 1 and 2.

Answer: B

gmatophobia · Answer

2x^2+9 < 9x

2x^2+ 9 - 9x< 0

The roots of the equation are  \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots are ⇒  \frac{9 \pm \sqrt{81 - 72}}{4}

\frac{9 \pm \sqrt{9}}{4}

Roots are ⇒  \frac{9 + 3}{4} = 3   ,  \frac{9 - 3}{4} = 1.5

Hence,  x  lies between  1.5  and  3 . As x is an integer, the only possible value of x =  2

2 has two positive factors.

Option B

Purnank · Answer

I think its A,

2x^2+9-9x  < 0

after solving we will get (2x - 3)*(x-3) < 0

now substitute +ve values of x starting from 1 and check for inequality you will see only x = 2 will be valid for inequality.

Hence A.

Nidzo · Answer

2x^2+9<9x

2x^2 - 9x + 9 < 0

2x^2 - 6x - 3x + 9 < 0

2x(x-3)(-3)(x-3) < 0

(2x - 3)(x-3)<0

x < \frac{3}{2}  and  x < 3

As we are dealing solely with positive integers of  x , the range of positive integers for which the inequality will hold is:  1.5 < x < 3 . The only positive integer which fits this range is  2 .

2  has two positive factors:  1  and  2 .

ANSWER B

chetan2u · Answer

The colored portion is correct but the question does not ask you for possible values of x. It asks you the positive factors of x, which is 2 in this case. 
Positive factors of 2 are 1 and 2, so answer is B.

prantorboni · Answer

If, x=1, we got, 11<9 which is wrong
X=2, 17<18 which seems perfect
X=3, 27<27 which is wrong
X=4, 41<36 which is wrong as well, so the raise in value of x continues to prove the equation wrong in the same pattern.

Certainly, 2 is the value of x for this equation. Since, 2 is a prime number, it has only two positive factors, 1 & 2 itself.
So, the answer is B here.

MBAApplicant2026 · Answer

Hi, I’m confused about this. If the 2 solutions are x is less than 3 and x is less than 1.5, then why is the solution range showing x can be greater than 1.5? Doesn’t the more restrictive solution (x being less than 1.5) take over so that the only solution is 1?

Posted from my mobile device

Bunuel · Answer

That solution is not precise. Check correct one here: https://gmatclub.com/forum/if-x-is-an-i ... l#p3404068 Hope it helps.

Kinshook · Answer

If  x  is an integer and  2x^2+9 < 9x , how many positive factors does  x  have?

2x^2 -9x +9 <0

2x^2 -6x -3x +9 <0

(2x -3)(x-3) < 0

-1.5 < x < 3

x = 2

Has factors 1 & 2

IMO B

If x is an integer and 2x^2 + 9 < 9x, how many positive factors

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GMAT Problem Solving (PS) Questions

If x is an integer and 2x^2 + 9 < 9x, how many positive factors

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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