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If 4<(7-x)/3, which of the following must be true?

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If 4<(7-x)/3, which of the following must be true? [#permalink]

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Problem Solving
Question: 156
Category: Algebra Inequalities
Page: 82
Difficulty: 600


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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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SOLUTION

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive


(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Note that we are asked to determine which MUST be true, not could be true.

\(4<\frac{7-x}{3}\) --> \(12<7-x\) --> \(x<-5\). So we know that \(x<-5\), it's given as a fact. Now, taking this info we should find out which of the following inequalities will be true OR which of the following inequalities will be true for the range \(x<-5\).

Basically the question asks: if \(x<-5\) which of the following is true?

I. \(5<x\) --> not true as \(x<-5\).

II. \(|x+3|>2\), this inequality holds true for 2 cases, (for 2 ranges): 1. when \(x+3>2\), so when \(x>-1\) or 2. when \(-x-3>2\), so when \(x<-5\). We are given that second range is true (\(x<-5\)), so this inequality holds true.

Or another way: ANY \(x\) from the range \(x<-5\) (-5.1, -6, -7, ...) will make \(|x+3|>2\) true, so as \(x<-5\), then \(|x+3|>2\) is always true.

III. \(-(x+5)>0\) --> \(x<-5\) --> true.

Answer: D.
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III



Sol:The given expression can be 4<(7-x)/3 (multiply by 3) we get
12<7-x or x<-5

St 1 can be ruled out

St 3 must be true So Option A,C and E can be ruled out

Consider st 2 |x+3|>2

When x+3 >/ 0 then |x+3|= x+3 and also x>/-3

or x+3> 2 or x>-1 but our condition is x>/-3 so x> -1 (because if x>-1 then x surely is greater than -3)

When x+3 <0 then |x+3|=- (x+3) and the st2 equation becomes x+3<-2 or x<-5 and thus x<-5 (because of x<-5 then x <-3 as well)

So we have x >-1 or x<-5 ------> In this range the St 2 will hold good. But we are given that x<-5 thus St 2 is true

Ans is D....
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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Solved the equation to get x < -5
As the question says must be true, it means that for any below -5 the statements have to hold true. So just assumed it to be -6 in every statement

Answer D
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 12 Mar 2014, 07:42
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Answer is D
As statement 1 can be ruled out as from question statementx<-5
statement 2 holds ok as modulus of lx+3l>2 : as from question statementx<-5
staement 3 is also correct as from question statementx<-5 therefore multiplication of 2 negative term is positive
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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Bunuel wrote:
SOLUTION

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive


(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Note that we are asked to determine which MUST be true, not could be true.

\(4<\frac{7-x}{3}\) --> \(12<7-x\) --> \(x<-5\). So we know that \(x<-5\), it's given as a fact. Now, taking this info we should find out which of the following inequalities will be true OR which of the following inequalities will be true for the range \(x<-5\).

Basically the question asks: if \(x<-5\) which of the following is true?

I. \(5<x\) --> not true as \(x<-5\).

II. \(|x+3|>2\), this inequality holds true for 2 cases, (for 2 ranges): 1. when \(x+3>2\), so when \(x>-1\) or 2. when \(-x-3>2\), so when \(x<-5\). We are given that second range is true (\(x<-5\)), so this inequality holds true.

Or another way: ANY \(x\) from the range \(x<-5\) (-5.1, -6, -7, ...) will make \(|x+3|>2\) true, so as \(x<-5\), then \(|x+3|>2\) is always true.

III. \(-(x+5)>0\) --> \(x<-5\) --> true.

Answer: D.




Bunuel,

For Statement 2, the red part does not hold true as X>-1 but green part holds true. So in such situation when one part of inequality is true, does it qualify for " Must be true"?
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 04 Jan 2015, 04:49
Bunuel wrote:
SOLUTION

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive


(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Note that we are asked to determine which MUST be true, not could be true.

\(4<\frac{7-x}{3}\) --> \(12<7-x\) --> \(x<-5\). So we know that \(x<-5\), it's given as a fact. Now, taking this info we should find out which of the following inequalities will be true OR which of the following inequalities will be true for the range \(x<-5\).

Basically the question asks: if \(x<-5\) which of the following is true?

I. \(5<x\) --> not true as \(x<-5\).

II. \(|x+3|>2\), this inequality holds true for 2 cases, (for 2 ranges): 1. when \(x+3>2\), so when \(x>-1\) or 2. when \(-x-3>2\), so when \(x<-5\). We are given that second range is true (\(x<-5\)), so this inequality holds true.

Or another way: ANY \(x\) from the range \(x<-5\) (-5.1, -6, -7, ...) will make \(|x+3|>2\) true, so as \(x<-5\), then \(|x+3|>2\) is always true.

III. \(-(x+5)>0\) --> \(x<-5\) --> true.

Answer: D.






For (2):
X could be a positive number or a negative number, if x is a negative number than solving inequality would result into X < -5 which is true because it is the exact same inequality which has been given in the question.

But if x would be positive that solving inequality would result into x >-1 which would make the option (2) not true. So in other words when x is negative than statement (2) is true but when x is positive than statement (2) is not true. So statement (2) is not ALWAYS true.


Can you please tell me why im unable to get the right answer? what is flaw in my reasoning?
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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mulhinmjavid wrote:
Bunuel wrote:
SOLUTION

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive


(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Note that we are asked to determine which MUST be true, not could be true.

\(4<\frac{7-x}{3}\) --> \(12<7-x\) --> \(x<-5\). So we know that \(x<-5\), it's given as a fact. Now, taking this info we should find out which of the following inequalities will be true OR which of the following inequalities will be true for the range \(x<-5\).

Basically the question asks: if \(x<-5\) which of the following is true?

I. \(5<x\) --> not true as \(x<-5\).

II. \(|x+3|>2\), this inequality holds true for 2 cases, (for 2 ranges): 1. when \(x+3>2\), so when \(x>-1\) or 2. when \(-x-3>2\), so when \(x<-5\). We are given that second range is true (\(x<-5\)), so this inequality holds true.

Or another way: ANY \(x\) from the range \(x<-5\) (-5.1, -6, -7, ...) will make \(|x+3|>2\) true, so as \(x<-5\), then \(|x+3|>2\) is always true.

III. \(-(x+5)>0\) --> \(x<-5\) --> true.

Answer: D.






For (2):
X could be a positive number or a negative number, if x is a negative number than solving inequality would result into X < -5 which is true because it is the exact same inequality which has been given in the question.

But if x would be positive that solving inequality would result into x >-1 which would make the option (2) not true. So in other words when x is negative than statement (2) is true but when x is positive than statement (2) is not true. So statement (2) is not ALWAYS true.


Can you please tell me why im unable to get the right answer? what is flaw in my reasoning?


It should be other way around.

To elaborate more. Question uses the same logic as in the examples below:

If \(x=5\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x>-10

Answer is E (x>-10), because as x=5 then it's more than -10.

Or:
If \(-1<x<10\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x<120

Again answer is E, because ANY \(x\) from \(-1<x<10\) will be less than 120 so it's always true about the number from this range to say that it's less than 120.

Or:
If \(-1<x<0\) or \(x>1\), then which of the following must be true about \(x\):
A. x>1
B. x>-1
C. |x|<1
D. |x|=1
E. |x|^2>1

As \(-1<x<0\) or \(x>1\) then ANY \(x\) from these ranges would satisfy \(x>-1\). So B is always true.

\(x\) could be for example -1/2, -3/4, or 10 but no matter what \(x\) actually is it's IN ANY CASE more than -1. So we can say about \(x\) that it's more than -1.

On the other hand for example A is not always true as it says that \(x>1\), which is not always true as \(x\) could be -1/2 and -1/2 is not more than 1.

Hope it's clear.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 14 May 2015, 00:03
Bunuel wrote:
mulhinmjavid wrote:
Bunuel wrote:
SOLUTION

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive


(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Note that we are asked to determine which MUST be true, not could be true.

\(4<\frac{7-x}{3}\) --> \(12<7-x\) --> \(x<-5\). So we know that \(x<-5\), it's given as a fact. Now, taking this info we should find out which of the following inequalities will be true OR which of the following inequalities will be true for the range \(x<-5\).

Basically the question asks: if \(x<-5\) which of the following is true?

I. \(5<x\) --> not true as \(x<-5\).

II. \(|x+3|>2\), this inequality holds true for 2 cases, (for 2 ranges): 1. when \(x+3>2\), so when \(x>-1\) or 2. when \(-x-3>2\), so when \(x<-5\). We are given that second range is true (\(x<-5\)), so this inequality holds true.

Or another way: ANY \(x\) from the range \(x<-5\) (-5.1, -6, -7, ...) will make \(|x+3|>2\) true, so as \(x<-5\), then \(|x+3|>2\) is always true.

III. \(-(x+5)>0\) --> \(x<-5\) --> true.

Answer: D.






For (2):
X could be a positive number or a negative number, if x is a negative number than solving inequality would result into X < -5 which is true because it is the exact same inequality which has been given in the question.

But if x would be positive that solving inequality would result into x >-1 which would make the option (2) not true. So in other words when x is negative than statement (2) is true but when x is positive than statement (2) is not true. So statement (2) is not ALWAYS true.


Can you please tell me why im unable to get the right answer? what is flaw in my reasoning?


It should be other way around.

To elaborate more. Question uses the same logic as in the examples below:

If \(x=5\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x>-10

Answer is E (x>-10), because as x=5 then it's more than -10.

Or:
If \(-1<x<10\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x<120

Again answer is E, because ANY \(x\) from \(-1<x<10\) will be less than 120 so it's always true about the number from this range to say that it's less than 120.

Or:
If \(-1<x<0\) or \(x>1\), then which of the following must be true about \(x\):
A. x>1
B. x>-1
C. |x|<1
D. |x|=1
E. |x|^2>1

As \(-1<x<0\) or \(x>1\) then ANY \(x\) from these ranges would satisfy \(x>-1\). So B is always true.

\(x\) could be for example -1/2, -3/4, or 10 but no matter what \(x\) actually is it's IN ANY CASE more than -1. So we can say about \(x\) that it's more than -1.

On the other hand for example A is not always true as it says that \(x>1\), which is not always true as \(x\) could be -1/2 and -1/2 is not more than 1.

Hope it's clear.



I still don't understand the application of this, to the question. Sure, I understand that if x = 5 its > -10.

But, here, we get two different cases. Solving the question stem we get x < -5.

In "II" we have that x is either x < -5 or x > -1. If I plug in a value where x > -1. Say x = 3. The equation doesn't hold?

We get 4 < (7-3)/3 --> 12 < 4. Not true?

How can this be a valid answer choice then

Edit: Don't know if this is how you should go about doing it, correct me if I'm wrong:

If we have \(x < -5.\)

Plugging in a value < -5, say -6 into II: \(|-6+3| > 2\). This holds true, aka sufficient
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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Quote:
I still don't understand the application of this, to the question. Sure, I understand that if x = 5 its > -10.

But, here, we get two different cases. Solving the question stem we get x < -5.

In "II" we have that x is either x < -5 or x > -1. If I plug in a value where x > -1. Say x = 3. The equation doesn't hold?

We get 4 < (7-3)/3 --> 12 < 4. Not true?

How can this be a valid answer choice then

Edit: Don't know if this is how you should go about doing it, correct me if I'm wrong:

If we have \(x < -5.\)

Plugging in a value < -5, say -6 into II: \(|-6+3| > 2\). This holds true, aka sufficient


Hi erikvm,

Let me explain the solution to you using the representation on number line. We need to evaluate if for every value in the range of the question statement i.e. x < -5, do the statements hold true.

Refer the below diagram for the range given to us in the question & the other statements:

Image


Question Statement: The question statement gives us the range as x < -5. We need to see for every value of x < -5, are the inequality in the statements true.

Statement-I: St-I gives us the range as x > 5. We see that this range does not hold true for any value of x. Hence, it can't be true.

Statement-II: St-II gives us the range as x > -1 or x < -5. We see that for every value of x < -5 (i.e. the inequality in the question statement), we can write |x +3| > 2 holds. Hence this is a must be true statement.

Statement-III: St-III gives us the range as x < -5 which is nothing but the range in question statement. Hence this is a must be true statement.

Therefore, st-II & III are true for every value of x < -5 and hence must be true statements.

Hope its clear :-D

Regards
Harsh
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 19 May 2015, 05:12
Thanks, I'm just abit confused about the whole "must be true". I know how absolute values work, but \(|x+3| > 2\) provides me with 2 different values for x, so I just assumed that this statement was not true because we have two different possibilities for X
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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erikvm wrote:
Thanks, I'm just abit confused about the whole "must be true". I know how absolute values work, but \(|x+3| > 2\) provides me with 2 different values for x, so I just assumed that this statement was not true because we have two different possibilities for X


Actually, \(|x+3| > 2\) provides you infinite values of x. It gives you two ranges: x < -5 OR x > -1.
The inequality implies that distance of x from -3 is more than 2. So x can lie to the right of -1 or to the left of -5. (Draw it on a number line to see). Hence you get two ranges: x < -5 OR x > -1

The question gives you that x < -5. So x will take any value less than -5. When it does, for all such values \(|x+3| > 2\) will always be true because it holds for all values such that x < -5. It holds for some more values too (x > -1) but we don't care about those. All values that x can take in our question, for those \(|x+3| > 2\) will always work. So \(|x+3| > 2\) must be true.
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 16 Mar 2016, 22:40
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Problem Solving
Question: 156
Category: Algebra Inequalities
Page: 82
Difficulty: 600


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The answer is B. Statement II is not always true. The question is about 'must be' true.
What is given is x<-5. There are not two answers. Statement II should produce a unique answer that satisfies x<-5.
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 16 Mar 2016, 22:56
VeritasPrepKarishma wrote:
erikvm wrote:
Thanks, I'm just abit confused about the whole "must be true". I know how absolute values work, but \(|x+3| > 2\) provides me with 2 different values for x, so I just assumed that this statement was not true because we have two different possibilities for X


Actually, \(|x+3| > 2\) provides you infinite values of x. It gives you two ranges: x < -5 OR x > -1.
The inequality implies that distance of x from -3 is more than 2. So x can lie to the right of -1 or to the left of -5. (Draw it on a number line to see). Hence you get two ranges: x < -5 OR x > -1

The question gives you that x < -5. So x will take any value less than -5. When it does, for all such values \(|x+3| > 2\) will always be true because it holds for all values such that x < -5. It holds for some more values too (x > -1) but we don't care about those. All values that x can take in our question, for those \(|x+3| > 2\) will always work. So \(|x+3| > 2\) must be true.

It is true, you need not care if the question is framed like 'which of the following is true for all values of x?'.
Otherwise, you need to care. The problem is confirmation bias.
Fallacy 1: Framing
Fallacy 2: Appeal to Authority
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 04 May 2016, 05:26
Why can statement 1 not be true? if we consider x to be negative, the question stem gives us x>5 as a possible result of x.

Thank you.
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 04 May 2016, 05:31
BobbyAssassinCross wrote:
Why can statement 1 not be true? if we consider x to be negative, the question stem gives us x>5 as a possible result of x.

Thank you.



Hi BobbyAssassinCross,
look at the highlighted portion..
initially you start with x as -ive and then get a +ive value for x as x>5..
so contradictory solution..
hence NOT possible
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 02 Jun 2016, 07:42
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Problem Solving
Question: 156
Category: Algebra Inequalities
Page: 82
Difficulty: 600


GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
1. Please provide your solutions to the questions;
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!


4<(7-x)/3
12<(7-x)
5<-x
x<-5

That means x is -ve and less than -5
Now lets look at the options

I. 5<x not possible. we know that x<-5

II. |x+3|>2 putting any -ve value for x <-5 will yield +ve value >2. True

III. -(x+5) is positive. Since X <-5, expression (x-5) will be -ve and because it is multiplied by -ve, it will give +ve result. True

(D) II and III only is the answer
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 03 Jan 2018, 22:46
Hi Bunuel can we think of 2nd statement as a Big set of ( 0,1,2,3,4,5 and also -6,-7,-8...) and what we need for must be true is a subset (-6, -7, -8...) ?
Is this correct?
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If 4<(7-x)/3, which of the following must be true? [#permalink]

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Prashant10692 wrote:
Hi Bunuel can we think of 2nd statement as a Big set of ( 0,1,2,3,4,5 and also -6,-7,-8...) and what we need for must be true is a subset (-6, -7, -8...) ?
Is this correct?


We are not given that x is necessarily an integer.

For |x + 3| > 2 to hold true, x < -5 or x > -1 must be true.

We know that x < -5, so |x + 3| > 2, will be true.

Check other similar questions from Trickiest Inequality Questions Type: Confusing Ranges (part of our Special Questions Directory).
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Re: If 4<(7-x)/3, which of the following must be true? [#permalink]

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New post 06 Feb 2018, 14:48
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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III


First, let's deal with the given inequality.
4 < (7-x)/3
Multiply both sides by 3 to get: 12 < 7 - x
Add x to both sides: x + 12 < 7
Subtract 12 from both sides to get: x < -5

So, if x < -5, which of the following statements MUST be true?

Aside: When dealing with "MUST be true" questions, we can eliminate a statement if we can find an instance where it is not true.

I. 5 < x (MUST this be true?)
No!
If x < -5, then it could be the case that x = -7, and -7 is NOT greater than 5
So, statement I need NOT be true.

II. |x+3| > 2 (MUST this be true?)
The answer is Yes. Here's why:
IMPORTANT CONCEPT: |x - k| represents the DISTANCE between x and k on the number line.
So, for example, we can think of |4 - 7| as the distance between 4 and 7 on the number line.
Notice that |4 - 7| = |-3| = 3, and 3 is indeed the distance between 4 and 7 on the number line.

Now let's examine |x+3|
We can rewrite this as |x - (-3)|
This represents the DISTANCE between x and -3 on the number line.
So, the inequality |x-(-3)| > 2 is stating that the DISTANCE between x and -3 on the number line is GREATER THAN 2
Well, since we're told that x < -5, we can be certain that the DISTANCE between x and -3 on the number line is definitely GREATER THAN 2
[If you're not convinced, sketch a number line, and place a big dot at -3. Then choose ANY value for x such that x < -5. You'll see that the distance between x and -3 is greater than 2]
So, statement II MUST be true.

III. -(x+5) is positive
This is the same as saying -(x+5) > 0 (MUST this be true?)
The answer is Yes. Here's why:
We're told that x < -5
If we add 5 to both sides we get x+5 < 0
Now, if we multiply both sides by -1, we get -(x+5) > 0
[aside: notice that, since I multiplied both sides by a negative value, I reversed the direction of the inequality]
As we can see, statement III MUST be true.

Answer: D

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Brent
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