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# If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f

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Manager
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If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f  [#permalink]

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27 Nov 2018, 08:19
2
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Difficulty:

85% (hard)

Question Stats:

41% (02:30) correct 59% (01:47) wrong based on 51 sessions

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If x and y are non-negative integers, x+y<11, and x−y>8, which of the following must be true for all the qualified values of x?

A. y < 3
B. y > 2
C. 2 < y < 10
D. y < 2
E. y < 1
VP
Joined: 09 Mar 2016
Posts: 1284
Re: If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f  [#permalink]

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27 Nov 2018, 08:32
rencsee wrote:
If x and y are non-negative integers, x+y<11, and x−y>8, which of the following must be true for all the qualified values of x?

A. y < 3
B. y > 2
C. 2 < y < 10
D. y < 2
E. y < 1

IMO: E hopefully

x+y<11 (if y y < 3 , ie 2 then 8+2<10 but 8-2 is not > 8)

x−y>8

if y<1 i.e. 0 then 10+0<11 and 10-0>8
CEO
Joined: 12 Sep 2015
Posts: 3726
Re: If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f  [#permalink]

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27 Nov 2018, 09:05
1
Top Contributor
2
rencsee wrote:
If x and y are non-negative integers, x+y<11, and x−y>8, which of the following must be true for all the qualified values of x?

A. y < 3
B. y > 2
C. 2 < y < 10
D. y < 2
E. y < 1

KEY CONCEPT: If the inequality signs of two inequalities are facing the SAME DIRECTION, then we can ADD those inequalities to create a new inequality.

We have:
x + y < 11
x − y > 8

Multiply both sides of the bottom inequality by -1 to get:
x + y < 11
-x + y < -8 [since we multiplied both sides by a NEGATIVE value, we REVERSED the inequality sign]

Now we can ADD the inequalities to get: 2y < 3
Divide both sides by 2 to get: y < 1.5
So, if y < 1.5, which of the answer choices MUST be true?
Well, if y < 1.5, then it MUST be the case that y < 2

RELATED VIDEOS FROM OUR COURSE

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VP
Joined: 09 Mar 2016
Posts: 1284
If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f  [#permalink]

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27 Nov 2018, 09:11
GMATPrepNow wrote:
rencsee wrote:
If x and y are non-negative integers, x+y<11, and x−y>8, which of the following must be true for all the qualified values of x?

A. y < 3
B. y > 2
C. 2 < y < 10
D. y < 2
E. y < 1

KEY CONCEPT: If the inequality signs of two inequalities are facing the SAME DIRECTION, then we can ADD those inequalities to create a new inequality.

We have:
x + y < 11
x − y > 8

Multiply both sides of the bottom inequality by -1 to get:
x + y < 11
-x + y < -8 [since we multiplied both sides by a NEGATIVE value, we REVERSED the inequality sign]

Now we can ADD the inequalities to get: 2y < 3
Divide both sides by 2 to get: y < 1.5
So, if y < 1.5, which of the answer choices MUST be true?
Well, if y < 1.5, then it MUST be the case that y < 2

RELATED VIDEOS FROM OUR COURSE

GMATPrepNow hi there

why are you multiplying both sides by -1 ? whats the logic behind making things "complicated" changing this x − y > 8 into
-x + y < -8

we dont have inaequlity like this for example $$-5>x$$ so as to multiply both sides by -1, i simply didnt see the neccessity to do so

and finally whats wrong with my solition
Manager
Joined: 04 Oct 2017
Posts: 81
If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f  [#permalink]

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27 Nov 2018, 09:15
GMATPrepNow wrote:
rencsee wrote:
If x and y are non-negative integers, x+y<11, and x−y>8, which of the following must be true for all the qualified values of x?

A. y < 3
B. y > 2
C. 2 < y < 10
D. y < 2
E. y < 1

KEY CONCEPT: If the inequality signs of two inequalities are facing the SAME DIRECTION, then we can ADD those inequalities to create a new inequality.

We have:
x + y < 11
x − y > 8

Multiply both sides of the bottom inequality by -1 to get:
x + y < 11
-x + y < -8 [since we multiplied both sides by a NEGATIVE value, we REVERSED the inequality sign]

Now we can ADD the inequalities to get: 2y < 3
Divide both sides by 2 to get: y < 1.5
So, if y < 1.5, which of the answer choices MUST be true?
Well, if y < 1.5, then it MUST be the case that y < 2

RELATED VIDEOS FROM OUR COURSE

Hi,

I subtracted the inequalities
x+ y <11
x-y>8

and got y< 1.5. Y is an integer hence y<2. Option D. Is my approach correct???
CEO
Joined: 12 Sep 2015
Posts: 3726
Re: If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f  [#permalink]

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27 Nov 2018, 09:27
1
Top Contributor
dave13 wrote:
GMATPrepNow hi there

why are you multiplying both sides by -1 ? whats the logic behind making things "complicated" changing this x − y > 8 into
-x + y < -8

we dont have inaequlity like this for example $$-5>x$$ so as to multiply both sides by -1, i simply didnt see the neccessity to do so

and finally whats wrong with my solition

Hi Dave,

Our goal is to combine the two inequalities so that the x terms are eliminated (this is our goal because the answer choices have y-terms only)
When it comes to combining two inequalities, one option is to ADD the inequalities. HOWEVER, we can only add the inequalities if the inequality symbols are facing the same direction.
In order to get the inequalities facing the same direction, I tool one inequality and multiplied both sides by -1

See the two embedded videos for more on these concepts.

I'm not entirely sure what you are doing in your solution. Can you elaborate on it for me?

Cheers,
Brent
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CEO
Joined: 12 Sep 2015
Posts: 3726
Re: If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f  [#permalink]

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27 Nov 2018, 09:36
Top Contributor
Kezia9 wrote:
GMATPrepNow wrote:
rencsee wrote:
If x and y are non-negative integers, x+y<11, and x−y>8, which of the following must be true for all the qualified values of x?

A. y < 3
B. y > 2
C. 2 < y < 10
D. y < 2
E. y < 1

Hi,

I subtracted the inequalities
x+ y <11
x-y>8

and got y< 1.5. Y is an integer hence y<2. Option D. Is my approach correct???

In this case, subtracting the inequalities worked, HOWEVER it's a bit of a coincidence (resulting from the fact that the inequality symbols were facing in opposite directions).

Also, when you subtracted the inequalities, you got 2y on one side, and 3 on the other side, but what rationale did you use to decide which direction to make the inequality symbol?
That is, why didn't you write 2y > 3?

IMPORTANT: In many cases, if you SUBTRACT inequalities, the resulting inequality will NOT be true.
For example, if we take...
x < 10
y < 4
...we can't conclude that x - y < 6
For example, it could be the case that x = 9 and y = -10, in which case x - y = 19, which contradicts the conclusion that x - y < 6

To be 100% certain of your solution, I suggest that you stick with the rules/strategies presented in the above videos.

Cheers,
Brent
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Test confidently with gmatprepnow.com
Re: If x and y are non-negative integers, x+y<11,andx−y>8 , which of the f   [#permalink] 27 Nov 2018, 09:36
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