If -2

Question

If  -2 < x \leq {5} , then which of the following MUST be true?

A)  -1 < x^2 < 16

B)  4 < x^2 \leq {25}

C)  0 \leq {x^2} \leq {16}

D)  0 < x^2 \leq {25}

E)  -4 \leq {x^2} < 36

BrentGMATPrepNow · Accepted Answer

For this kind of question, I usually test values. It's fast and helps avoid careless mistakes.

If -2 < x  <  5, then it could be the case that x = 5, in which case x² = 25
In other words, it's  possible  that  x² = 25 
Check the answer choices...
Answer choice A says x² < 16. In other words, it says that x² can't equal 25. ELIMINATE A
Answer choice C says x²  <  16. In other words, it says that x² can't equal 25. ELIMINATE C

Also, if -2 < x  <  5, then it could be the case that that x = 0, in which case x² = 0
In other words, it's  possible  that  x² = 0 
When we check the remaining answer choices, we see that...
Answer choice B (4 < x²) says x² can't equal 0. ELIMINATE B
Answer choice D (0 < x²) says x² can't equal 0. ELIMINATE D

We're left with 1 answer choice...E

EDIT: Given some of the discussions below, I thought I'd remind new visitors what it means for a number to be greater than another number.
x < y means  y lies to the right of x on the number line . That's it. 
Similarly, the inequality 0 < x tells that x lies to the right of 0 on the number line. 
Similarly, the inequality -4 < x² < 36 means x² lies to the right of -4 and to the left of 36 on the number line.
Finally, the inequality -4 ≤ x² < 36 means x² EITHER equals -4 (which it certainly doesn't)  OR  x² lies to the right of -4 on the number line (which it certainly does),  and  x² lies to the left of 36 on the number line (which it certainly does)

In other words, saying -4 ≤ x² < 36 doesn't imply that x² could equal -2. 
Similarly, everyone agrees that -4 < 1, but that doesn't imply that 1 could equal -2.

Kurtosis · Answer

Given:  -2 < x \leq {5} 
Value of  x^2  can range from 0 to 25.
Only option E covers the above range

Answer: E

Alternate method:
Put x = 5 --> A and C can't be true
Put x = 0 --> B and D can't be true

Choice E works for all x between -2 and 5.

Answer: E

afarrukh786 · Answer

Square of any number can never b negative, answer D is correct.

Bunuel · Answer

From the stem x CAN be 0, and if it is, then D is wrong. The correct answer is E. Please re-read the solutions above.

afarrukh786 · Answer

From the stem x CAN be 0, and if it is, then D is wrong. The correct answer is E. Please re-read the solutions

D option does not say equal to zero, it says greater than zero, if you see option D, there is not a sign of equal to

ashishahujasham · Answer

I could agree with all of you and eliminate abcd but what about e square of any number cannot be negative so how is option e true.

Can anyone tell why option E.

Sent from my A1601 using  GMAT Club Forum mobile app

adamm412 · Answer

This to me is a simple question that is complicated by GMAT trickery in the way the answer choices are presented. the smallest value for x^2 is 0, and 0 is always greater than -4, so option E still fulfills the demands of the question. It is always true that x^2 is greater than or equal to -4 because it is always greater than -4.

afarrukh786 · Answer

How can E be answer while it include negative numbers and zero also

adamm412 · Answer

Again it comes down to the way the question is presented: "which of the following MUST be true". In theory, answer E could have been -1,000,000 ≤ x^2 < 1,000,000 and it would still be the only correct answer because it is the only solution offered that contains the actual possible values of x^2. notice the maximum value of x^2 is 25 but answer E allows for up to 36... we know x^2 cannot get that big, but the way the solution is presented means x^2 never actually needs to reach ~36 in order for it to still satisfy those constraints. This is the same idea on the negative lower bound of x^2.

Also, x^2 = 0 is a possible solution if x=0.

Vinayak Shenoy · Answer

Ok lets make it simpler. Given is  -2 < x \leq {5} 
which means x can have the following values -1, 0, 1, 2, 3, 4, 5 (for simplification purpose i'm using integers. x can have a decimal values also as question doesn't say x is an integer)
now see what all values can x^2 take. x^2 can take 0, 1, 4, 9, 16, 25. Yes?
with this knowledge approach the answer choice. Only option E gives you all the above values of x^2.

Bunuel · Answer

I think you don't understand the logic of the question. The question asks: if  -2 < x \leq {5} , then which of the following MUST be true? So, we KNOW that  -2 < x \leq {5}  and should evaluate which of the options must be correct. If  -2 < x \leq {5} , then only E must be true.

chetan2u · Answer

Hi,
No calculations required, just logic on the choices given...

E has the widest range and contains ALL other choices, so if any of A to D is correct, E has to be correct....
But can we have TWO answers, No..
So E is the answer

BrentGMATPrepNow · Answer

Many students will assume that E is incorrect, because x² CANNOT equal -4. However, answer choice E does not suggest that x² can equal -4, it only states that x² must be GREATER THAN or equal to -4.

Here's a similar example: If Joe has more than 5 shirts, which of the following can we conclude with certainty?
A) Joe owns more than 6 shirts.
NO. We can't conclude this, because it's possible that Joe owns exactly 6 shirts, and 6 is not greater than 6.

B) Joe owns more than -3 shirts.
YES. If Joe owns more than 5 shirts, then he definitely owns more than -3 shirts 
Does this mean that Joe COULD own -2 shirts? No. If just means that we can be certain that he owns more than -3 shirts.

Does that help?

Cheers,
Brent

akankshasehgal16 · Answer

I have to disagree with the solution here. E tells us that |x| is less that 6, which could imply x=5.99999 and so on. However the range is fixed from -2 to 5, and x is not necessarily an integer. If it was, I would have agreed with the solution (E).

For me, I choose D.

For me, I choose D.

chetan2u · Answer

The range is -2<x≤5

least value for x^2 is 0 when x is 0.
Maximum value for x^2 is 25 when x is 5.

So the actual range is 0 ≤ x ≤ 25.

Now D does not contain 0.

Only E contains this range

monkinaferrari · Answer

The question stem didn't say that we have to include all the values, it just says "must be true".

E's range starts from -4 to 36.

All the values fall within this range but we have extra values as well.

D's range starts from 0 to 25. This is 100% true. Every value falls within this range.

E makes less sense to me.

Rickooreoisb · Answer

How come x be < 36, x cannot take value more than 25, which means x cannot be 30 and hence E must be eliminated

Bunuel · Answer

E doesn’t say x^2 can be 30. It just says every possible x^2 must stay below 36. And 0 to 25 already stay below 36, so E is satisfied.

P.S. This question belongs to a type of question that tends to be very confusing for many. Therefore, I recommend practicing similar questions from the following collection:  Trickiest Inequality Questions Type: Confusing Ranges .

If -2 < x <= 5, then which of the following MUST be true?

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