Inequality to represent minimum monthly income.

[Richie Smash]

Homework Statement

I Type I Cost per pupil I
I Full session I $50 I
I Half session I $30 I

The above table shows the cost of lessons per month to students attending a private class.
The class operates under the following limitations:
1. The maximum number of students in the class is 40.
2. There must be a minimum of 5 Full Session students.
3. The number of half session Students must be at least half the number of full session students.
4.The minimum monthly income must be $1200

Let X represent the number of full session students, and let Y represent the number of half session students.
Hence state four inequalities, not including x ≥ 0 and y ≥0 to represent the conditions above.

Homework Equations

The Attempt at a Solution

So for the first one, I know that it would be x+y≤40.
The second one 5≤x
The third one 1/2x≤y
But the fourth one... I am kinda stuck here... that one is a bit more difficult
My best guess would be 5x+1/2x≥$1200...

 

[Mark44]

Homework Statement

I Type I Cost per pupil I
I Full session I $50 I
I Half session I $30 I

The above table shows the cost of lessons per month to students attending a private class.
The class operates under the following limitations:
1. The maximum number of students in the class is 40.
2. There must be a minimum of 5 Full Session students.
3. The number of half session Students must be at least half the number of full session students.
4.The minimum monthly income must be $1200

Let X represent the number of full session students, and let Y represent the number of half session students.
Hence state four inequalities, not including x ≥ 0 and y ≥0 to represent the conditions above.

Homework Equations

The Attempt at a Solution

So for the first one, I know that it would be x+y≤40.
The second one 5≤x
The third one 1/2x≤y
But the fourth one... I am kinda stuck here... that one is a bit more difficult
My best guess would be 5x+1/2x≥$1200...

The first three inequalities are fine.
For the fourth, you have x full-session students and y half-session students. How much per month does it cost for one full-session student and how much for one half-session student?
From your inequality these numbers appear to be $5 and $.50 respectively.

 

[Richie Smash]

For the fourth, you have x full-session students and y half-session students. How much per month does it cost for one full-session student and how much for one half-session student?
From your inequality these numbers appear to be $5 and $.50 respectively.

Oh I didn't intend for that to be the money, I used the following information to deduce that, if the minimum of full session students must be 5, then it would be 5x.
And since the minimum of half session students is 1/2x, it would be actually 2.5x.

What I'm trying to do is say, using the minium amount of students for full sessions, and the minimum of students for half sessions, the income generated must be greater than or equal to $1200. I am trying to represent that statement of which I understand the concept, I don't know how to turn that into inequality.

Wait a second, It just came to me, x represents the number of full session, and y the number of half session students, If I say something like 5x, that means that I'm saying 5 times however many full session students there are.

But we need the money, I just thought of something, it's $50 per full session, $30 per half session,

So I could say $50x +$30y≥$1200?

and my book has 5x+3y≥120 as the answer.. I am very close I can sense that.

 

[Mark44]

Oh I didn't intend for that to be the money, I used the following information to deduce that, if the minimum of full session students must be 5, then it would be 5x.
And since the minimum of half session students is 1/2x, it would be actually 2.5x.

Your 2nd and 3rd inequalities already represent these relationships.
The fourth inequality is intended to be about the revenue that x full-session students and y half-session students bring in.

What I'm trying to do is say, using the minium amount of students for full sessions, and the minimum of students for half sessions, the income generated must be greater than or equal to $1200.

Don't make things more complicated than they need to be. Each full-session student brings in $50 and each half-session student brings in $30. And you know what the minimum monthly revenue needs to be.

 

[Richie Smash]

Hi mark, I was wondering if you could confirm my answer of $50x +$30y≥$1200 is correct, but why does my book have 5x+3y≥120

 

[Mark44]

Hi mark, I was wondering if you could confirm my answer of $50x +$30y≥$1200 is correct, but why does my book have 5x+3y≥120

Yes, yours is correct, and so is the book's answer. Just divide both sides of your inequality by 10 to the book's inequality. One of the properties of both equations and inequalities is that you can divide both sides by a positive value and get an equivalent inequality. (If you divide both sides of an inequality by a negative number, the direction of the inequality changes.)

 

[Ray Vickson]

Oh I didn't intend for that to be the money, I used the following information to deduce that, if the minimum of full session students must be 5, then it would be 5x.
And since the minimum of half session students is 1/2x, it would be actually 2.5x.

What I'm trying to do is say, using the minium amount of students for full sessions, and the minimum of students for half sessions, the income generated must be greater than or equal to $1200. I am trying to represent that statement of which I understand the concept, I don't know how to turn that into inequality.

Wait a second, It just came to me, x represents the number of full session, and y the number of half session students, If I say something like 5x, that means that I'm saying 5 times however many full session students there are.

But we need the money, I just thought of something, it's $50 per full session, $30 per half session,

So I could say $50x +$30y≥$1200?

and my book has 5x+3y≥120 as the answer.. I am very close I can sense that.

Right, but remove the $ signs. If you submitted your inequalities to a computer and asked it to describe feasible region, it would choke, and give you an error message because it would not know how to interpret "$".