# Doubt on inequality question

**URL:** <https://talk.achievable.me/t/doubt-on-inequality-question/1809>\
**Category:** Quantitative reasoning\
**Created:** [August 2, 2022, 1:45pm UTC](https://talk.achievable.me/t/doubt-on-inequality-question/1809 "2022-08-02T13:45:23Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![Colonial\_olive\_marmo](https://talk.achievable.me/letter_avatar/colonial_olive_marmo/32/6_449c533f586939b6e5f246632a46421f.png) [@Colonial\_olive\_marmo](https://talk.achievable.me/u/Colonial_olive_marmo)\
**Post date:** [August 2, 2022, 1:45pm UTC](https://talk.achievable.me/t/doubt-on-inequality-question/1809/1 "2022-08-02T13:45:23Z")

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Hello, I came across a question TY9C6-RQKVH while solving inequalities. Here is the question.

If 5x + 5 \>55 and xy \< 0 then which of the following must be true if y is an integer?

y+1 \< 2  
5y \> y  
y-4 \< -7  
y^2 \< 0  
y \> x

Explanation . If we simplify for x we get x\>10. Also, xy \< 0 . So y\<0

Right choice y + 1 \< 2. This means y \< 1. Which means y could take values 0 and below on the number line. When y=0 , xy=0 and xy \< 0 condition fails.

I chose option y-4\< -7 . I wanted to know why this is wrong and the choice y+1 \<2 is correct.

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**Author:** ![Justin](https://talk.achievable.me/user_avatar/talk.achievable.me/justin/32/41_2.png) [@Justin](https://talk.achievable.me/u/Justin)\
**Post date:** [August 2, 2022, 4:37pm UTC](https://talk.achievable.me/t/doubt-on-inequality-question/1809/2 "2022-08-02T16:37:56Z")

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Hi @Colonial_olive_marmo, your logic is very close!

Once we have _y\<0_, you can basically just plug into the equations and see what would work for different versions of _y_.

The choice _y-4\<-7_ works in some cases, but is not always correct, because rearranged, this choice can be expressed as _y\<-3_, which does not completely overlap with _y\<0_. You can see this if you plug in _y=-1_. You would get _(-1)-4 \< -7_, which simplifies to _-5 \< -7_, which is not true.

The choice _y+1\<2_ can be rearranged to _y\<1_, which does overlap completely with _y\<0_. Plugging in _y=-1_ does work, as you get _(-1)+1 \< 2_, which simplifies to _0\<2_, which is true. Even if you use a more negative number like _y=-100_, the choice is still true: _(-100) + 1 \< 2_ simplifies to _-99 \< 2_, which is still true.

Please let me know if I can clarify further!

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**Author:** ![Colonial\_olive\_marmo](https://talk.achievable.me/letter_avatar/colonial_olive_marmo/32/6_449c533f586939b6e5f246632a46421f.png) [@Colonial\_olive\_marmo](https://talk.achievable.me/u/Colonial_olive_marmo)\
**Post date:** [August 14, 2022, 3:49pm UTC](https://talk.achievable.me/t/doubt-on-inequality-question/1809/3 "2022-08-14T15:49:40Z")

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Hello Justin,

I agree with your description. However, I feel you don’t consider the corner case when y=0.

y+1 \< 2 implies y \< 1 and not y\< 0. Since y\< 1 , y can take values left of 1 , that includes 0. The issue with choosing y=0 is then it invalidates xy\< 0. So, I still don’t feel y+1\< 2 is correct. On the other hand if it was y+3\< 2 or any other combination where, after simplification we get y\<0, I would have no doubt in my mind.

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**Author:** ![Justin](https://talk.achievable.me/user_avatar/talk.achievable.me/justin/32/41_2.png) [@Justin](https://talk.achievable.me/u/Justin)\
**Post date:** [August 14, 2022, 6:04pm UTC](https://talk.achievable.me/t/doubt-on-inequality-question/1809/4 "2022-08-14T18:04:39Z")

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Hi @Colonial_olive_marmo,

We’ll never be able to have _y = 0_ because we already determined from the given equations that we must have _y \< 0_ in all cases. It isn’t a possibility for it to be zero or greater because it would violate the constraints of the question.

As _y_ is also defined to be an integer, the greatest it could be is _y = -1_

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**Author:** ![Colonial\_olive\_marmo](https://talk.achievable.me/letter_avatar/colonial_olive_marmo/32/6_449c533f586939b6e5f246632a46421f.png) [@Colonial\_olive\_marmo](https://talk.achievable.me/u/Colonial_olive_marmo)\
**Post date:** [August 16, 2022, 12:06am UTC](https://talk.achievable.me/t/doubt-on-inequality-question/1809/5 "2022-08-16T00:06:53Z")

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I see your point now. With xy\< 0 and x\> 10, we know that y \<0 . We are looking for an answer that gives us all ranges of y. So y+1\< 2 is correct answer compared to y-4\<-7. Thank you @Justin