Is x 0? (1) 1/x x

Question

Is x > 0?

(1) 1/x < 1

(2) 1/x > x

Source: Experts' Global

Nups1324 · Answer

Hi Maxximus I checked the video explanation but I have a doubt. Why can't we multiply by x on both sides and thus get 1x^2 from statement.? Why do we need to only test numbers as shown in the video explanation? Thank you

Deepakjhamb · Answer

Put numbers and solve

U will get always no in case if c

Posted from my mobile device

Maxximus · Answer

Hello,

Multiplying or dividing by a variable is not a valid operation in inequations; because if the variable is negative, it may alter the inequality.

Example:

3 > 2

but it doesn't mean that 3x > 2x always holds

For negative values of x,

3x < 2x

This is one of the earliest concepts explained in our inequations videos; please revise the concept videos.

All the best!
Experts' Global

Nups1324 · Answer

Oh. Okay I got my mistake. I only considered x as positive, x can be negative as well. I subconsciously took x as positive after seeing the question. My bad. Apologies. Now it all makes sense. Thank you

ravigupta2912 · Answer

st 1 --> 1/x < 1 => 1 < x OR x < 0 st. 2 --> 1/x > x => 1 > x > 0 OR x < -1 Merge the two statements and the only common point is that x can be negative.

ScottTargetTestPrep · Answer

First of all, testing numbers is not the only way to solve this question. The reason we wouldn’t multiply each side of 1/x < 1 by x is that we have no information on the sign of x. If x were positive, then we would obtain x > 1, as you noticed, but if x were negative, we would obtain x < 1 since we have to change the direction of an inequality whenever we multiply each side by a negative number. To solve 1/x < 1, move the 1 on the right hand side to the left hand side: 1/x - 1 < 0 Now get the common denominator x and then combine the two fractions: 1/x - x/x < 0 (1 - x)/x < 0 Now, there are two ways this expression can be negative: Case 1 : 1 - x is negative and x is positive, or Case 2 : 1 - x is positive and x is negative. For Case 1 , we get 1 - x < 0 (which is equivalent to x > 1) and x > 0. The set of numbers that satisfy both of the inequalities x > 0 and x > 1 are x > 1. So, one way to satisfy the inequality 1/x < 1 is to pick values for x which are greater than 1. For Case 2 , we get 1 - x > 0 (which is equivalent to x < 1) and x < 0. The set of numbers that satisfy both inequalities are the set of x such that x < 0. Thus, the inequality 1/x < 1 can also be satisfied by picking a negative value for x. As we can see, x could be either greater than 1 or less than 0. If we were to multiply each side of 1/x < 1 by x, we would completely miss the fact that negative values of x also satisfy the inequality 1/x < 1. The explanation of why it’s wrong to rewrite 1/x > x as x^2 < 1 is similar to the explanation just presented.

Louis14 · Answer

Statement#1: Not Sufficient

1/x < 1
1/x - 1 < 0
1-x/x < 0
(x-1)/x > 0 (x can be positive or negative)

Statement#2: Not Sufficient

1/x > x
1/x - x > 0
1 - x^2/x > 0
x^2 - 1/x < 0
(x+1)(x-1)/x < 0 (x can be positive or negative)

Combined 1&2:

Statement#1 tells us that (x-1)/x > 0
If we put this in statement#2 i.e (x+1)(x-1)/x < 0, then it implies that x+1<0 because (x-1)/x > 0.
Hence, x<-1.

If x < -1, then it is definitely not greater than 0. Hence, we have a definite NO.

Sufficient. Answer (C)

CEdward · Answer

For statement 2, if we choose a positive integer, say, 3, wouldn't that violate the inequality? i.e. isn't the LHS supposed to be negative?

Fdambro294 · Answer

Is x > 0 ?

Or

Is X = (+)Positive Value?

(1) 1/X < 1

Case 1: X = Pos.

X >1 ------ YES

Case 2: X = Neg.

X < 1 ----- and Since X < 0 b/c we are assuming it is Negative?

the Limiting Inequality stands as: X < 0 ------NO

S1 NOT SUFFICIENT

(2) 1/x > x

Testing Values in the 4 Ranges:

X > +1

0 < X < +1

-1 < X < 0

X < -1

the Ranges of X that Satisfy the Inequality are:

X < -1 ----- NO

or

0 < X < 1 ------YES

S2 NOT SUFFICIENT ALONE

TOGETHER:

If X is Positive, it must Satisfy BOTH the Inequalities in Statement 1 and Statement 2:

S1: if X is Positive: X > 1

S2: 0 < X < 1

since it is impossible for a Positive Value to Satisfy BOTH Ranges at the same time, X can NOT be a Positive Value

S1: if X is Negative: X < 1

thus, S1 tells us that any Value in the Range: X < 0 satisfies S1's Inequality

S2: X < -1

therefore, any Value in the Range of: X < -1

will Satisfy Both Statements at the Same Time

X must be negative. Definite NO.

(C)Together Sufficient

Louis14 · Answer

Sure, 3, a positive number, may violate the inequality, but what if x= 1/2, which is also a positive number? In that case (x=1/2), x can assume a positive value without violating the inequality. Ultimately, even if one and only one value satisfies an inequality, the value stands valid. This means that statement 2 is still insufficient. Hope it helps.

Deyoz · Answer

i tried in this way-

State 1: 1/X < 1 = (1-x)/x < 0;
x-x^2 < 0 or x < X^2

this implies that X could either positive or negative, so Not sufficient.

State II:

1/X > X;
1/X-X>0;
1-X^2/X > 0;
(X-X^3) > 0;
X > X^3;
inference; either X is negative or X is a fraction. so so Statement II also both Yes and no therefore is sufficient.

However combining Statement I and II, we can say that X is negative. (statement i says X either positive or negative) (statement 2 says, either negative or fraction), hence X must be negative.

So answer C

rgoel2305 · Answer

what i did:

to check for C,

1) 1/x < 1
2) x < 1/x

Adding both, 1/x + x < 1 + 1/x

=> subtracting 1/x from both sides, x < 1.

So, answer should be E, but i know this is wrong because values like x = 0.5 doesnt satisfy St.1.

Can anyone explain what am i doing wrong when i am adding and trying to solve them algebraically?

Thanks in advance.

Bunuel

RenB · Answer

Solved it by trying cases and drawing inferences from the statements

St 1: 
X can be negative - Not satisfied
X can be positive - Satisfied
2 possibilities- one satisfies the condition, other doesnt, thus not sufficient

St 2: 
X can be negative - Not satisfied
X can be a value between 0 and 1 (This is a catch)- Satisfied
2 possibilities- one satisfies the condition, other doesnt, thus not sufficient

St 1+ St 2: 
Common and the only possibility. X is negative - Not satisfied
Only one solution with a single result- Sufficient

gmatchile1 · Answer

What happens with: Is x>3? 1)x>2 2)x<2 Help! https://avisodato.cl/listing/clases-gma ... nalizadas/

Bunuel · Answer

Technically, statement (2) would be sufficient, since it gives a definite “No” answer to the question. However, this DS question is flawed because on the GMAT the two statements never directly contradict each other, while here (1) and (2) clearly do.

P.S. Pure algebraic questions are no longer a part of the  DS syllabus  of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

Word Problems 
 Work Problems 
 Distance Problems 
 Mixture Problems 
 Percent and Interest Problems 
 Overlapping Sets Problems 
 Statistics Problems 
 Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check  GMAT Syllabus for Focus Edition

You can also visit the  Data Sufficiency forum  and filter questions by  OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources  to see the types of questions currently tested on the GMAT.

So, you can ignore this question. 
 
Hope it helps.

gmatchile1 · Answer

Thank you,  Bunuel

For your selfless help.
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