Inequalities

[newton9]

Is x/3 + 3/x > 2?

(1) x < 3

(2) x > 1

[Brent@GMATPrepNow]

Is x/3 + 3/x > 2?

(1) x < 3

(2) x > 1

Statement 1
We can show this is insufficient through counter-example.
If x < 3, then:
a) x could equal 1, in which case x/3 + 3/x IS greater than 2
b) x could equal -1, in which case x/3 + 3/x IS NOT greater than 2
Since statement 1 yields 2 different answers to the target question, it is not sufficient

Statement 2
If x > 1, then x must be positive.
This is very useful, because if we know that x is positive, we can take our target question "Is x/3 + 3/x > 2?" and multiply both sides by 3x to get a new target question " Is x^2 + 9 > 6x?"
If we take our new target question and subtract 6x from both sides, we get "Is x^2 - 6x + 9 > 0?
Finally, if we factor the left hand side, we get "Is (x - 3)^2 > 0?"
Well, (x - 3)^2 is almost always greater than zero. The only time it is not greater than zero is when x = 3.
As such, statement 2 is not sufficient.

Statements 1 & 2 combined
From statement 2, we reworded the target question as Is (x - 3)^2 > 0?
Statement 1 tell us that x cannot equal 3
This means that (x - 3)^2 must ALWAYS be greater than 0
As such, the two statements combined are sufficient, and the answer is C

[arunpanda22]

Is x/3 + 3/x > 2?

(1) x < 3

(2) x > 1

Statement 1
We can show this is insufficient through counter-example.
If x < 3, then:
a) x could equal 1, in which case x/3 + 3/x IS greater than 2
b) x could equal -1, in which case x/3 + 3/x IS NOT greater than 2
Since statement 1 yields 2 different answers to the target question, it is not sufficient

Statement 2
If x > 1, then x must be positive.
This is very useful, because if we know that x is positive, we can take our target question "Is x/3 + 3/x > 2?" and multiply both sides by 3x to get a new target question " Is x^2 + 9 > 6x?"
If we take our new target question and subtract 6x from both sides, we get "Is x^2 - 6x + 9 > 0?
Finally, if we factor the left hand side, we get "Is (x - 3)^2 > 0?"
Well, (x - 3)^2 is almost always greater than zero. The only time it is not greater than zero is when x = 3.
As such, statement 2 is not sufficient.

Statements 1 & 2 combined
From statement 2, we reworded the target question as Is (x - 3)^2 > 0?
Statement 1 tell us that x cannot equal 3
This means that (x - 3)^2 must ALWAYS be greater than 0
As such, the two statements combined are sufficient, and the answer is C

why not do it this way
check if (x-3)^2>0 on condition 1
if x<3 then (x-3)^2 will be greater than 0
so answer should be a

[Brent@GMATPrepNow]

why not do it this way
check if (x-3)^2>0 on condition 1
if x<3 then (x-3)^2 will be greater than 0
so answer should be a

Excellent question!

For statement 1, it would be great if we could take the target question "Is x/3 + 3/x > 2?" and simplify it by multiplying both sides by 3x (as I did for statement 2).
The problem is that, before we can multiply (or divide) any inequality by a variable (in this case 3x), we must be absolutely certain of the sign (positive/negative) of that variable.

There's an important rule concerning inequalities that essentially tells us that:

If we multiply or divide both sides of an inequality by a positive number, then the inequality sign stays as it is. If we multiply or divide both sides of an inequality by a negative number, then the inequality sign must be reversed.

Example: If we take the inequality 3 < 5 and multiply both sides by 2 (positive), we leave the inequality sign as it is to get 6 < 10
Conversely, if we take the inequality 3 < 5 and multiply both sides by -2 (negative), then we must reverse the inequality sign to get -6 > -10

Now consider the inequality in the target question: Is x/3 + 3/x > 2?
Statement 1 says that x < 3. Now if we multiply both sides of the inequality by 3x, it is unclear whether 3x is positive or negative. We only know that x < 3, which means 3x can be either positive or negative.
Since it is unclear whether 3x is positive or negative, then we don't know if the inequality sign should stay as it is or be reversed. As such, we can't perform this operation.

However, when it comes to statement 2, (x>1) we can be certain that 3x is positive, which means we can multiply both sides of the inequality by 3x (which must be positive) and we will leave the inequality sign as it is.

I hope that helps.

Cheers,
Brent

[arunpanda22]

why not do it this way
check if (x-3)^2>0 on condition 1
if x<3 then (x-3)^2 will be greater than 0
so answer should be a

Excellent question!

For statement 1, it would be great if we could take the target question "Is x/3 + 3/x > 2?" and simplify it by multiplying both sides by 3x (as I did for statement 2).
The problem is that, before we can multiply (or divide) any inequality by a variable (in this case 3x), we must be absolutely certain of the sign (positive/negative) of that variable.

There's an important rule concerning inequalities that essentially tells us that:

If we multiply or divide both sides of an inequality by a positive number, then the inequality sign stays as it is. If we multiply or divide both sides of an inequality by a negative number, then the inequality sign must be reversed.

Example: If we take the inequality 3 < 5 and multiply both sides by 2 (positive), we leave the inequality sign as it is to get 6 < 10
Conversely, if we take the inequality 3 < 5 and multiply both sides by -2 (negative), then we must reverse the inequality sign to get -6 > -10

Now consider the inequality in the target question: Is x/3 + 3/x > 2?
Statement 1 says that x < 3. Now if we multiply both sides of the inequality by 3x, it is unclear whether 3x is positive or negative. We only know that x < 3, which means 3x can be either positive or negative.
Since it is unclear whether 3x is positive or negative, then we don't know if the inequality sign should stay as it is or be reversed. As such, we can't perform this operation.

However, when it comes to statement 2, (x>1) we can be certain that 3x is positive, which means we can multiply both sides of the inequality by 3x (which must be positive) and we will leave the inequality sign as it is.

I hope that helps.

Cheers,
Brent

thanks for clearing up
gr8 explanation