Is −3 ≤ x ≤ 3 ? (1) x^2 + y^2 = 9 (2) x^2 + y ≤ 9

Question

Is  −3 ≤ x ≤ 3  ?

(1)  x^2 + y^2 = 9 
(2)  x^2 + y ≤ 9

DS28402.01

Bunuel · Accepted Answer

APPRACH 1: Is −3 ≤ x ≤ 3 ? (1) x^2 + y^2 = 9 If x were greater than 3 or less than -3, then x^2 would be greater than 9. Since y^2 \geq 0 , then x^2 + y^2 would be greater than 9, and not equal to it. Therefore, x cannot be greater than 3 or less than -3, which means that -3 \leq x \leq 3 . Sufficient. (2) x^2 + y \leq 9 Rearranging the inequality, we get x^2 ≤ 9 - y . However, since there is no constraint on the value of y, the statement is just saying that x^2 is less than or equal to some unknown number (the statement might as well be x^2 ≤ z, where we know nothing about z), which is clearly insufficient. Not sufficient. APPRACH 2: Is −3 ≤ x ≤ 3 ? (1) x^2 + y^2 = 9 The equation x^2 + y^2 = 9 represents a circle centered at the origin with a radius of 3 . Therefore, the x-coordinate of any point on the circle must be between -3 and 3 . Hence, the answer is -3 \leq x \leq 3 . https://gmatclub.com/forum/download/file.php?id=70408 Sufficient. (2) x^2 + y \leq 9 The inequality y \leq -x^2 + 9 represents the region below a downward-facing parabola that opens towards the negative y-axis. The vertex of this parabola is at the origin and its arms extend indefinitely. Therefore, x can take any value, so this statement is also insufficient. https://gmatclub.com/forum/download/file.php?id=70409 Not sufficient. download.gif download (1).gif

gvij2017 · Answer

Statement 1 
Any value of x beyond   will make the expression x^2 + y^2 >9 because y^2 is always >=0. 
Sufficient.

Statement 2 
Take x =4 in expression x^2+y<=9 
For many values of y, expression can hold. 
take x= 0, expression can still hold for other values of y. 
Not sufficient.

A is answer.

nick1816 · Answer

Statement 1- x^2+y^2=9 x^2=9-y^2 As y^2≥0 0≤x^2≤9 -3≤x≤3 Sufficient Statement 2- x^2+y≤9 y ≤ 9-x^2 Capture.PNG x∈ Insufficient

BrentGMATPrepNow · Answer

Target question :    Is -3 ≤ x ≤ 3?

Statement 1 : x² + y² = 9  
First we need to recognize that x² and y² are  greater than or equal to 0  for all possible values of x and y
Considering this property, let's find the minimum and maximum possible values of x
In order to maximize the value of x we must minimize the value of y²
The minimum value of y² occurs when y = 0. 
When y = 0, we get: x² + 0² = 9, which is the same as x² = 9, which means x = 3 or x = -3
For all other values of y, x will be BETWEEN 3 and -3
So, the answer to the target question is  YES, it IS the case that -3 ≤ x ≤ 3 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : x² + y ≤ 9 
There are several values of x and y that satisfy statement 2. Here are two: 
 Case a : x = 0 and y = 0. In this case, the answer to the target question is  YES, it IS the case that -3 ≤ x ≤ 3 
 Case b : x = 4 and y = -10. In this case, the answer to the target question is  NO, it is NOT the case that -3 ≤ x ≤ 3 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Niha01 · Answer

Can someone please explain the approach to solve this question? please

Jitu20 · Answer

Niha01

Statement 1: x^2 + y^2 = 9 => y^2 = \sqrt{9 - x^2}

Since root of negative number is not possible, sufficient to answer −3 ≤ x ≤ 3

Statement 2: A range of values are possible as this is an inequality and we do not know exact values of x or y. i.e. no unique value.

chetan2u · Answer

Nothing much in the question stem except a range for x. But let us simplify it. 
 −3 ≤ x ≤ 3  =  | x |≤ 3  or  x^2\leq{9}

(1)  x^2 + y^2 = 9 
 y^2\geq{0} , as all squares are non-negative. 
 x^2=9-y^2   That is, the value of x^2 decreases from 9 to 0 as value of y^2 increases from 0 to 9  
So  {x^2}\leq 9  always irrespective of value of y. 
Sufficient

(2)  x^2 + y ≤ 9 
 x^2\leq 9-y 
So  x^2  will be greater than 9, when y is negative, but will be lesser than 9 when y is positive. 
Insufficient

A

achloes · Answer

Bunuel   chetan2u

For statement 1, can't we just square both sides of the equation?

By doing so, we'd be left with... x + y = 3

In which case...

0+3= 3 
1+2 = 3
2+1 =3
3+0 = 3

All satisfy the question, therefore the statement is sufficient.

Please let me know if this approach is acceptable - thanks!

Bunuel · Answer

I think you mean taking the square root rather than squaring. The problem with that is that  \sqrt{x^2 + y^2}  generally does not equal to x + y (it does when on of the unknowns is 0 and another is nonzero). Fore example,  \sqrt{3^2 + 4^2}=5 , while 3 + 4 = 7.

Moreover, even if (1) were x + y = 3, it would be insufficient. Any value of x will satisfy it when y = 3 - x. For example, if x = -100, then y = 103.

I believe you intended to say  take the square  root instead of  square  . However, this presents a problem as  \sqrt{x^2 + y^2}  does not typically equal x + y (with the exception of one variable being zero and the other nonnegative). For instance,  \sqrt{3^2 + 4^2}=5 , while 3 + 4 = 7.

Furthermore, even if we assume that (1) is x + y = 3, it would still be insufficient. Any value of x could satisfy this equation as long as y = 3 - x. For instance, if x = -100, then y would equal 103.

Hope it's clear.

yashas123 · Answer

Can anyone help me with my approach in this question?

I think that we take these type of questions as a yes or no question like yes x lies in this range or no x doesn't lie in this range.

So for the second statement as we can see that different values of x satisfies this equation ranging from x=0 to x<-3 also. So my approach for answering this question was like since x can also take values which lies outside the given range in question , so no x doesn't lie in this range. Hence the second st is sufficient to answer this question.

Bunuel · Answer

If x = y = 0, then  x^2 + y ≤ 9  is satisfied, and -3 ≤ x ≤ 3, giving us a YES answer to the question.

If x = 4 and y = -100, then  x^2 + y ≤ 9  is also satisfied, but x > 3, giving us a NO answer.

Since we get different answers (YES and NO), the second statement is not sufficient. You can check a more detailed analysis of the question here:  https://gmatclub.com/forum/is-3-x-3-1-x-2-y-2-9-2-x-2-y-305950.html#p3163372

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.

yashas123 · Answer

Thanks for the doubt clearance and specially with the syllabus updation thing.

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Is −3 ≤ x ≤ 3 ? (1) x^2 + y^2 = 9 (2) x^2 + y ≤ 9

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GMAT Data Sufficiency (DS) Questions

Is −3 ≤ x ≤ 3 ? (1) x^2 + y^2 = 9 (2) x^2 + y ≤ 9

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