Is it true that x > 0?

st.1
\(x^2=2x\)
\(x^2-2x=0\)
\(x(x-2)=0\)

\(x=0\) or \(x=2\) . For x=0 answer to the question is no and for x=2, answer is yes. hence st.1 alone is not sufficient

st.2
\(x^3=3x\)
\(x(x^2-3)=0\)
\(x(x-\sqrt{3})(x+\sqrt{3})=0\)
\(x=0\), \(x=\sqrt{3}\) or \(x= -\sqrt{3}\)
clearly both yes and no answer to the original question are possible hence not sufficient

st.1 + st.2

the only common solution for both of these equation is x=0, which results in the answer to the original question as no.

thus answer should be C

x^2 = 2x

We can solve the equation as follows:

x^2 - 2x = 0

x(x - 2) = 0

x = 0 or x = 2

We see that if x = 0, then x is not greater than 0. However, if x = 2, then x is greater than 0. Statement one alone is not sufficient.

Statement Two Alone:

x^3 = 3x

We can solve the equation as follows:

x^3 - 3x = 0

x(x^2 - 3) = 0

x = 0 or x^2 = 3
x = 0 or x = √3 or x = -√3

We see that if x = 0 or -√3, then x is not greater than 0. However, if x = √3, then x is greater than 0. Statement two alone is not sufficient.

Statements One and Two Together:

Using both statements, we see that x can only be 0. Thus, x is not greater than 0. Both statements together are sufficient.

Answer: C
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why can't x2 = 2x be re-written as x multiplied by x= 2 multiplied by x and then cancel x with x an thus the answer is x=2 and thus A is sufficient?

You cannot reduce x^2 = 2x by x because x can be 0 and we cannot divide by 0. By doing so you loose a root, namely x = 0.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.

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Hi,

This is one of the popular traps in GMAT, that is, Cancelling of common variables on both sides.

Never fall for such a trap, i.e., do not cancel common variables on both sides of an equation unless it is already mentioned in the question as variables(unknowns) not equal to zero.

Statement I is insufficient:

x^2 = 2x

x^2 – 2x = 0

x(x-2) = 0

x and x-2 are two numbers, for the product to be zero not necessary that both of them have to be zero.

Either x = 0 or x-2 = 0

i.e.,

x=0 or x =2

If x = 0, then answer to the question is NO.

If x = 2, then answer to the question is YES.

Two answers are there, so not sufficient.

Here, easily most of the students end up cancelling “x” on both sides and say x =2, which is wrong to do, as x can be equal to 0.

Statement II is insufficient:

x^3 = 3x

Similar to statement I,

x^3 – 3x = 0

x(x^2-3) = 0

So, either x = 0 or x^2 -3 = 0

i.e., x = 0 or x = +/- (root 3).

Again we will have yes or no answer to the question.

So not sufficient.

Both statements together,

x=0 is the only value which satisfy both statements.

So answer has to be C(Together Sufficient).

While practicing learn from your mistakes.

Hope this helps.

Target question: Is it true that x > 0?

Statement 1: x² = 2x
Rewrite as: x² - 2x = 0
Factor: x(x - 2) = 0
So, EITHER x = 0 OR x = 2
Let's examine each possible case
Case a: If x = 0, then the answer to the target question is NO, it is not true that x > 0
Case b: If x = 2, then the answer to the target question is YES, it is true that x > 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x³ = 3x
Rewrite as: x³ - 3x = 0
Factor: x(x² - 3) = 0
Factor again: x(x - √3)(x + √3) = 0
So, x = 0, OR x = √3 OR x = -√3
Let's examine each possible case
Case a: If x = 0, then the answer to the target question is NO, it is not true that x > 0
Case b: If x = √3, then the answer to the target question is YES, it is true that x > 0
Case c: If x = -√3,, then the answer to the target question is NO, it is not true that x > 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x = 0 OR x = 2
Statement 2 tells us that x = 0, OR x = √3 OR x = -√3
The only x-value that satisfies BOTH statements is x = 0
So, the answer to the target question is NO, it is not true that x > 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

#1
x(x-2)=0
x=0,+2
insufficient
#2
x(x^2-3)=0
x=0 and x=+/-√3
insufficient
from 1 & 2
x=0
sufficeint
IMO C

Is it true that x > 0?

(1) x^2 = 2x
(2) x^3 = 3x

(1) x^2=2x ---> x(x-2)=0---> x=0 or x=2. Not Sufficient
(2) for the same logic as (1), (2) is Not Sufficient

(1)+(2) ---> x=3/2>0.

this problem involves rearranging the statements
1. x^2-2x=0
x=0 or x=2
NS
2. x^3-3x=0
x=0 or x=+/-1.7
NS
combined
x=0 is the only overlap and both get me an answer of no so the answer is C

Bunuel

From both statement 1 and 2 we are getting x=0 and x= some positive value also i.e. x>0 (2 and 1.73) respectively
Since both statements give solutions as x=0 and x>0, shouldnt the answer be E in this case

Many thanks for your help
Devansh

From (1) x can be 0 or 2.
From (2) x can be 0, √3 or -√3.

When taken together x can only be 0, because x = 2 does not satisfy (2) and x = √3 or -√3 does not satisfy (1). The only common value to satisfy both statements is x = 0.

Hope it's clear.
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When two statements are combined does that mean the answer values of both statements or does that mean, \(x^{2}-2x=x^{3}-3x\) ?

I am really confused, in some DS questions, while selecting option C, we combine the two statements to derive/solve and see that they both are helpful to get an answer whether a yes/no or particular answer. but in this answer explanation, you have taken the value that satisfies from statement 1 and S2..

Can you combine those two statements and derive as to how option C is correct ?

No! when you are combining two statements, that doesn't mean you are equating them
It simply means, you are using the results or info in those statements in whichever way logically possible to arrive at a UNIQUE answer. Here 'a UNIQUE answer' is very important to understand. There may be a situation where equating 2 information stems would help you arrive at a unique solution and a situation where it might not help. It is entirely dependent on the nature of the question.

Coming back to this question - Even I made the error when I even overlooked the fact that we always need to arrive at a UNIQUE solution in DS while that/those solution(s) must also make sense with each statement. Why solution(s) must make sense with both statements? Because, Statements 1 and 2 are considered true or deemed true statements i.e. contradicting those statements would mean contradicting with question stem.
Hence, as explained by Bunuel - both numbers 2 and root 3 and -ve root 3 wont satisfy the other statement(s), consequently it would be incorrect to say that we arrived at a UNIQUE answer which contradicts one of the statements.

Best, Devansh

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