M28-46

Question

Is  x^2 \gt 2x ?

(1)  x  is a prime number.
 
(2)  x^2  is a multiple of 9.

Bunuel · Answer

Official Solution:

Is  x(x-2) > 0 ? 
 
 x(x-2) > 0  is true when  x < 0  or  x > 2 . Essentially, if  x  is not 0, 1, or 2, we have a YES answer to the question.
 
(1)  x  is a prime number. 
 
If  x=2 , then the answer is NO, but if  x  is some other prime, then the answer is YES. Not sufficient.
 
(2)  x^2  is a multiple of 9. 
 
If  x=0 , then the answer is NO, but if  x=3 , then the answer is YES. Not sufficient.
 
(1)+(2) Since from (1),  x  is a prime, and from (2),  x^2  is a multiple of 9, then  x  can only be 3. Therefore, the answer to the question is YES. Sufficient.

Answer: C

patilvrishali101 · Answer

Hi Bunuel,

Can we consider Is x2>2x? as 'IS X>2'?

Thanks.

Bunuel · Answer

No. We cannot reduce an equation by a variable if we don't know its sign: if x is positive, then yes, from x^2 > 2x we get x > 2 BUT if x is negative, then when reducing by negative value we must flip the sign and we'd get x < 2. For more check below links: Inequalities Made Easy! Solving Quadratic Inequalities - Graphic Approach Inequality tips DS Inequalities Problems PS Inequalities Problems 700+ Inequalities problems inequalities-trick-91482.html data-suff-inequalities-109078.html range-for-variable-x-in-a-given-inequality-109468.html everything-is-less-than-zero-108884.html graphic-approach-to-problems-with-inequalities-68037.html

stonecold · Answer

Excellent Question ..
here i was tempted to choose A and B but here is why i chose C
here in statement 1 => x can be 2 
in statement 2 x can be 0 too as 0 is the multiple of every number .
Hence C

Celestial09 · Answer

Hi!
A small doubt..
x(x-2)>0
x>2 and x,0...
but is it not x>2 and x> 0?
why inequity sign changed in second while comparing to zero?
thanks
any imp theory link you like to share please...

Mo2men · Answer

I'm happy to answer your doubt, regardless of the question. I will answer it as we solve inequality. What does it mean X(X-2)>0 ? Let's put it in simple way. We have TWO number A & B are multiplied and result Bigger than 0. When does Happen? From NUMBER PROPERTIES: Case 1 : A = positive & B= positive ............ A * B >0 Case 2 : A =negative & B= negative ............ A * B >0 Let look to our inequality : X(X-2)>0 Put: A=X & B= X-2 Using cases mentioned above let's analyze our inequality: Case 1 : X (X-2)>0.......... Remember A & B positive means A >0 & B>0 X>0 & X-2>0 then X>2 ....... Can you come up with example that satisfy that X>0 & X>2 in same time???Answer:YES we can. Any number BIGGER than 2 does so CASE 1 yields that X>2 Case 2 X (X-2)>0.......... Remember A & B negative means A <0 & B<0 X<0 & X-2<0 then X<2 ........Can you come up with example that satisfy that X<0 & X<2 in same time???Answer:YES we can. Any number SMALLER than 0 does so. CASE 2 yields that X<0 From above Cases we can conclude FINALLY that: TO SATISFY the Inequality: Either X<0 or X>2 ................NOTICE that when we combine TWO CASES, we use 'Either......OR......' You can check number if you are in doubt. I will expand my explanation about how to solve if you face this question: X (X-2)<0 From NUMBER PROPERTIES: Case 1 : A = positive & B= negative ............ A * B <0 Case 2 : A =negative & B= positive ............ A * B <0 Put: A=X & B= X-2 Using cases mentioned above let's analyze our inequality: Case 1 : X (X-2)<0.......... Remember A = positive means A >0 & B=negative means B<0 X>0 & X-2<0 then X<2 ....... Can you come up with example that satisfy that X>0 & X<2 in same time???Answer:YES we can. Any number BIGGER than 0 & SMALLER than 2. We can put mathematically as 00.......... Remember A = negative means A < 0 & B=positive means B>0 X<0 & X-2>0 then X>2 ........Can you come up with example that satisfy that X<0 & X>2 in same time ???Answer:NO, we can NOT. It means that we need a number that is negative and positive in same time. Can you find it?

CASE 2 yields no solution From above Cases we can conclude FINALLY that: TO SATISFY the Inequality: 0

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M28-46

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