Inequality

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BTGmoderatorRO: Moderator; Posts: 772; Joined: Wed Aug 30, 2017 6:29 pmFollowed by: 6 members

by BTGmoderatorRO » Sun Oct 08, 2017 11:28 am

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
QA is d. what is the answer to this question and why is option C wrong

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Source: Beat The GMAT — Problem Solving | Sun Oct 08, 2017 11:28 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Oct 08, 2017 12:11 pm

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

I. 5<x
2. |x+3|>2
3. -(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

Simplify the expression in the question stem:
4 < (7-x)/3
12 < 7-x
x < -5.
Question stem rephrased:
If x < -5, which of the following must be true?

I: x > 5
Since x is negative, it is not possible that x>5.
Eliminate C and E, which include I.

II: |x+3| > 2.
|x-(-3)| > 2
|a-b| = the DISTANCE between a and b.
Thus, |x-(-3)| > 2 implies the following:
The distance between x and -3 is greater than 2.
Since x<-5, it must be true that x is more than 2 places from -3.
Eliminate B, which does not include II.

III: -(x+5)>0
x+5 < 0
x < -5.
Eliminate A, which does not include III.

The correct answer is D.

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Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Mon Oct 09, 2017 6:30 am

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

I. 5<x
2. |x+3|>2
3. -(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

Simplify the expression in the question stem:
4 < (7-x)/3
12 < 7-x
x < -5.
Question stem rephrased:
If x < -5, which of the following must be true?

II: |x+3| > 2.
|x-(-3)| > 2
|a-b| = the DISTANCE between a and b.
Thus, |x-(-3)| > 2 implies the following:
The distance between x and -3 is greater than 2.
Since x<-5, it must be true that x is more than 2 places from -3.
Eliminate B, which does not include II.

Dear Mitch,

I solved II in another way

|x+3| > 2

x +3 > 2................x>-1
or
x +3 < -2 .............x <-5

so, x >-1 or x <-5

In your solution, you have ignored the red part, making II not must be true?

Where did I go worng here?

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rsarashi: Master | Next Rank: 500 Posts; Posts: 186; Joined: Sat Dec 24, 2016 12:38 amFollowed by: 3 members; Thanked: 5 times

by rsarashi » Mon Oct 09, 2017 9:51 am

Dear Mitch,

I solved II in another way

|x+3| > 2

x +3 > 2................x>-1
or
x +3 < -2 .............x <-5

so, x >-1 or x <-5

In your solution, you have ignored the red part, making II not must be true?

Where did I go worng here?

[/quote]

Hi GMATGuruNY ,

I also want to know the same.

If we solve this we get x>-1 or x<-5

and x<-5 is true.

Please explain.

Thanks

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Oct 09, 2017 10:34 am

Roland2rule wrote: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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Oct 09, 2017 12:11 pm

Mo2men wrote:Dear Mitch,

I solved II in another way

|x+3| > 2

x +3 > 2................x>-1
or
x +3 < -2 .............x <-5

so, x >-1 or x <-5

In your solution, you have ignored the red part, making II not must be true?

Where did I go worng here?

Your solution is fine.
Prompt: x<-5.
Statement II: x<-5 or x>-1
The portion in green is like a green box that contains every value less than -5 or greater than -1.
Question stem, rephrased:
Is every value represented by the inequality in red contained within the green box implied by Statement II?
YES.
Thus, Statement II must be true.

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