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If x and y are numbers such that x < - 10 and y > 6, which of the

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If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post Updated on: 13 Dec 2016, 23:58
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If x and y are numbers such that x < - 10 and y > 6, which of the following expressions is true?

I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32
II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0
III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

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

This is Question 3 for the e-GMAT Question Series on Absolute Value.

Provide your solution below. Kudos for participation. Happy Solving! :-D

Best Regards
The e-GMAT Team

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Originally posted by EgmatQuantExpert on 20 May 2015, 10:20.
Last edited by EgmatQuantExpert on 13 Dec 2016, 23:58, edited 2 times in total.
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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 22 May 2015, 07:17
3
3
Official Explanation

Correct Answer: D

First representing the points x and y on the number line: x < -10 and y > 6.

Then, since the expressions talk about |x+4|, |x-4|, |y+4| and |y-4|, representing these distances on the number line as well.

|x+4| represents the distance of x from -4 on the number line
|x-4| represents the distance of x from 4
|y+4| represents the distance of y from -4
|y-4| represents the distance of y from 4

Image

I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32

From the diagram it’s clear that:
|x+4| > 6
|y + 4| >10
|y – 4| > 2
|x – 4| 14

Adding all these we get Expression 1. Therefore, is true always.

II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0
From the diagram, we can see that |x – 4| - |x + 4| = 8
And, |y + 4| - | y – 4| = 8
Adding the 2 equations, we get:

|x – 4| + |y + 4| - | x + 4| - | y – 4| = 16

So, expression 2 is NOT TRUE.


III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

From the diagram, we can see that |x + 4| - |x – 4| = -8
And, |y + 4| - | y – 4| = 8
Adding the 2 equations, we get:

|x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

So, expression 3 is TRUE.
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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 20 May 2015, 20:47
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As stated, If x and y are numbers such that x < - 10 and y > 6,

if y > 6 , then
    | y + 4| = y + 4
    | y - 4 | = y - 4
if x < - 10 , then
    | x – 4 | = - ( x – 4 )
    | x + 4| = - ( x + 4 )

lets solve each
I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32
    -(x+4) + (y + 4) + (y - 4) - (x – 4) > 32
    -x -4 + y + 4 + y - 4 -x +4 > 32
    -2x +2y > 32
    -x + y > 16
    -(-10.1) + (6.1) > 16 ; True

II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0
    -(x-4) + (y + 4) - (-(x + 4)) - (y-4) < 0
    -x +4 + y + 4 +x -4 -y +4 < 0
    8 < 0 ; False

III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0
    -(x+4) + (y + 4) - (-(x – 4)) - (y - 4) = 0
    -x -4 + y + 4 +x -4 -y + 4 = 0
    0 = 0 ; True

Ans : D
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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 20 May 2015, 14:26
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Tried solving the question by substituting values for x and y.
Suppose if we sub. x = -11 and y = 7, and solve the three equations, It's found that statements (i) and (iii) satisfy. Hence D.
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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 21 May 2015, 11:51
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EgmatQuantExpert wrote:
If x and y are numbers such that x < - 10 and y > 6, which of the following expressions is true?

I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32
II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0
III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

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

This is Question 3 for the e-GMAT Question Series on Absolute Value.

Provide your solution below. Kudos for participation. The Official Answer and Explanation will be posted on 22nd May.

Till then, Happy Solving! :-D

Best Regards
The e-GMAT Team


How to solve this:

Plug in smart values for x and y. The question tells us that x<-10 and y>6. Therefore I take -11 = x and 7 = y.

I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32
Plug in: 7 + 11 + 3 + 15 = 36 > 32

Be careful now, normally once you have an answer that is true, it doesn't mean that it will be true for all the values of x and y. Here im pretty confident because i took the smallest (integer) numbers for x and y and the result is 36 which is a difference of 4 to 32. Even if you plug in smaller fractional numbers the statement will still be true.

II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0
Plug in: 15 + 11 - 7 - 3 > 0, therefore false.

No other plug ins are needed since one false value makes us eliminating it directly.

III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0
Plug in: 7 + 11 - 15 - 3 = 0, therefore true.

But is it true for any values of x and y? What about y = 8 and x = -12? Let's try:

Plug in #2: 8 + 12 - 16 - 4 = 0, obviously thats always true.

Answer D like Dragon :-)

Any helpful recommendation how to do this in 2 minutes?
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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 22 May 2015, 07:22
reto wrote:
Any helpful recommendation how to do this in 2 minutes?


Dear reto

The key to answering this question within 2 minutes is: the ability to represent |x+4|, |x-4|, |y+4|, |y-4| on the number line. Do make sure that you are comfortable with this visual sense of absolute value. :)

Best Regards

Japinder
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Articles and Question to reach Q51 | Question of the week

Must Read Articles
Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets



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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 26 May 2015, 12:15
EgmatQuantExpert wrote:
reto wrote:
Any helpful recommendation how to do this in 2 minutes?


Dear reto

The key to answering this question within 2 minutes is: the ability to represent |x+4|, |x-4|, |y+4|, |y-4| on the number line. Do make sure that you are comfortable with this visual sense of absolute value. :)

Best Regards

Japinder


To be honest, I feel more comfortable with plugging in. Just to set up the number line as you did in your example takes me too much time. And I don't feel comfortable with the number line, I assume that the error probability is bigger in my case by visualising the number line than plugging in.
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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 05 Sep 2015, 00:12
EgmatQuantExpert wrote:
If x and y are numbers such that x < - 10 and y > 6, which of the following expressions is true?

I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32
II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0
III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

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

This is Question 3 for the e-GMAT Question Series on Absolute Value.

Provide your solution below. Kudos for participation. Happy Solving! :-D

Best Regards
The e-GMAT Team

but how do you confirm about this |x – 4| - |x + 4| = 8
And, |y + 4| - | y – 4| = 8 ???
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Re: If x and y are numbers such that x < - 10 and y > 6, which of the [#permalink]

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New post 16 Jul 2018, 08:16
EgmatQuantExpert wrote:
Official Explanation

Correct Answer: D

First representing the points x and y on the number line: x < -10 and y > 6.

Then, since the expressions talk about |x+4|, |x-4|, |y+4| and |y-4|, representing these distances on the number line as well.

|x+4| represents the distance of x from -4 on the number line
|x-4| represents the distance of x from 4
|y+4| represents the distance of y from -4
|y-4| represents the distance of y from 4

Image

I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32

From the diagram it’s clear that:
|x+4| > 6
|y + 4| >10
|y – 4| > 2
|x – 4| 14

Adding all these we get Expression 1. Therefore, is true always.

II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0
From the diagram, we can see that |x – 4| - |x + 4| = 8
And, |y + 4| - | y – 4| = 8
Adding the 2 equations, we get:

|x – 4| + |y + 4| - | x + 4| - | y – 4| = 16

So, expression 2 is NOT TRUE.


III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

From the diagram, we can see that |x + 4| - |x – 4| = -8
And, |y + 4| - | y – 4| = 8
Adding the 2 equations, we get:

|x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

So, expression 3 is TRUE.



What if non integers are considered? Will you solution work for that as well?
Re: If x and y are numbers such that x < - 10 and y > 6, which of the   [#permalink] 16 Jul 2018, 08:16
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