Tricky DS from Grockit

[loveusonu]

Is x/3 + 3/x >2

(1) x<2
(2) x>1

[spoiler]OA: C ; IMO B[/spoiler]

Here is my solution to reach B:

x/3 + 3/x >2
Cross multiplying the left hand side, we get

(x^2 + 9) / 3x > 2

we know x^2 will always be positive and 9 is also positive, so numerator will always be positive and greater than 9 for sure.

Now if x is -ve then eqn is false as denominator will be -ve.
And if x is +ve then eqn will be true as denominator will be +ve and offcourse x^2 + 9 will be many times greater.

In short question is asking, is X -ve or +ve ?

(1) x<2 ---> x can be positive or it can be -ve as well.(checked for max value 2 as well and answer is coming YES too)
(2) x>1 --> This says X is always +ve , meaning eqn is true.

Hence went with B.

Where am I going wrong.

[BastiG]

My solution is C.



I restate the formula by multiplying it with 3 to:

x+9/x>2

First checking statement 1:

X=1
1+9>6
X=-1
-1-9<6

false

Second checking statement 2:

x=2
2+4,5 > 6
Attention: number is getting smaller --> further testing
X=3
3+3 > 6
false

Checking both statements together:
extreme cases already tested X=1 and X=2

Thus answer is C.

[BastiG]

Is x/3 + 3/x >2


In short question is asking, is X -ve or +ve ? --> wrong conclusion. This is only true for the first statement but not for second. There it is important which number you choose.

[loveusonu]

Is x/3 + 3/x >2


In short question is asking, is X -ve or +ve ? --> wrong conclusion. This is only true for the first statement but not for second. There it is important which number you choose.

does that mean I can't cross multiply in inequalities, same like we can't mlutiply\divide X uless known +ve or -ve??
Because Cross multiplying x/3 + 3/x >2 leads to a different conclusion(as stated above)

[BastiG]

Is x/3 + 3/x >2


In short question is asking, is X -ve or +ve ? --> wrong conclusion. This is only true for the first statement but not for second. There it is important which number you choose.

does that mean I can't cross multiply in inequalities, same like we can't mlutiply\divide X uless known +ve or -ve??
Because Cross multiplying x/3 + 3/x >2 leads to a different conclusion(as stated above)

No you can cross-multiply. But look at the formula:

(x^2 + 9) / 3x > 2

If you input x=3

then

(9+9)/9=2

If You input x=4

then

(16+9)/12=2,08

Thus the second statement is insufficient.

[kvcpk]

Is x/3 + 3/x >2

(1) x<2
(2) x>1

Let me share a simple concept here.

a + 1/a >=2 whenever, a is positive.
a+1/a = 2 when a=1 else a+1/a>2

Now,
in tis case, a=x/3
Hence x/3 + 1/(x/3) >=2 when x/3 is positive.
x/3 + 1/(x/3) =2 when x/3=1 -> x=3
Else, x/3 + 1/(x/3) >2 for all positive x.

Hope this helps!!