# Are these linear programming constraints correct?

The problem is: Beth works a maximum of $20$ hours/week programming computers and tutoring math. She receives $\$25$/hour for programming and$\$20$/hour for tutoring. She works between $3$ and $8$ hours/week programming, but always gives more time to tutoring. How many hours should she work at each job to maximize her income?

Let $x$ = # hours programming and $y$ = # hours tutoring.

My constraints are:

Total hours: $x + y ≤ 20$

Hours programming: $3 ≤ x ≤ 8$

Hours tutoring: $y > x$

Are these right?

-
Seems OK. I would be more comfortable with $y\ge x$, even though it goes against the usual meaning of "more". –  André Nicolas Dec 3 '12 at 20:40
But then y could be equal to x, and y is always greater. –  Someone Dec 3 '12 at 20:41
Sure. But if you are ultimately solving "graphically," and the relevant corner involves the line $y=x$, we probably would not reject that as an answer. –  André Nicolas Dec 3 '12 at 20:48
But if I'm solving graphically, I would use the vertices, no? So it wouldn't matter if the line was y ≥ x or y > x since it would be the same line. –  Someone Dec 3 '12 at 20:50
Don't worry about it. –  André Nicolas Dec 3 '12 at 20:52

It looks good, though "between" is a bit ambiguous. Sometimes, it is meant the way that you interpreted it, but sometimes, it is meant to indicate strict inequalities.

-
Yes, but that's not what the restriction says, because x and y do not have to add up, only if she has to maximize her income. –  Someone Dec 3 '12 at 20:39
I think it's meant the way I interpreted it; I wouldn't worry. –  Someone Dec 3 '12 at 20:40
You're correct. I've deleted it from my answer. –  Cameron Buie Dec 3 '12 at 20:40
Okay, sounds good. Thanks so much for your quick answer! –  Someone Dec 3 '12 at 20:40
If it were meant the other way, we actually couldn't maximize income, so I'm sure you appropriately interpreted it. –  Cameron Buie Dec 3 '12 at 20:41