# Inequality from GRE

So I was helping my gf go over some GRE math questions, and we came across a strange question.

Solve for x and y
x=2y
5x<y+7

I said that the answer would be

y<7/9 and x<14/9

this is the right answer, but she asked how that works. How do both the equality and inequality hold true at the same time? Shouldn't the answer be undetermined? I don't quite know how to answer the question. My opinion is that as long as both x and y are less than their respective inequality, then the equality will then hold.

Are there any more precise thoughts or theories regarding the subject?

## Answers and Replies

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Pengwuino
Gold Member
If that is the right answer, it simply means that x and y have to simultaneously satisfy the conditions that y<7/ and x<14/9 and x=2y. So in other words, x = -5 and y = -3 would not satisfy the equation but x = -5 and y = -2.5 will. Your idea is a bit backwards, the equality holding is a much stronger condition but in the end, one condition being being true doesn't mean other is true

Last edited:
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
The inequality 5x<y+7 corresponds to the half plane above the line y=5x-7. The solutions correspond to the portion of the line x=2y that lies in that half plane.

HallsofIvy
Science Advisor
Homework Helper
Since x= 2y, the second inequality becomes 5(2y)< y+ 7. 10y< y+ 7 so 9y< 7, y< 7/9.

I would be inclined to say that "y< 7/9, x< 14/9" is NOT the correct answer because that implies that any point (x, y) in the quadrant of R2 satisfying x< 14/9, y< 7/9, satisfies those. That is not the case. The correct answer is "y< 7/9, x= 2y". That is, the solution set is the ray x= 2y for all y< 7/9.

Since x= 2y, the second inequality becomes 5(2y)< y+ 7. 10y< y+ 7 so 9y< 7, y< 7/9.

I would be inclined to say that "y< 7/9, x< 14/9" is NOT the correct answer because that implies that any point (x, y) in the quadrant of R2 satisfying x< 14/9, y< 7/9, satisfies those. That is not the case. The correct answer is "y< 7/9, x= 2y". That is, the solution set is the ray x= 2y for all y< 7/9.
Ah yes, that sounds better to me too. Even though that's not a possible answer, it makes better mathematical sense, and I agree that it should in fact be the ray x= 2y for all y< 7/9.