Inequality to represent minimum monthly income.

In summary, the given table shows the cost of private class lessons per month, with a maximum of 40 students, a minimum of 5 full-session students, and half the number of full-session students for half-session students. To represent these conditions, the inequalities x+y≤40, x≥5, and y≥1/2x are used. To also meet the minimum monthly income of $1200, the inequality 50x+30y≥1200 is added.
  • #1
Richie Smash
293
15

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...
 
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  • #2
Richie Smash said:

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.
 
  • #3
Mark44 said:
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.
 
Last edited:
  • #4
Richie Smash said:
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.

Richie Smash said:
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.
 
  • #5
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
 
  • #6
Richie Smash said:
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.)
 
  • #7
Richie Smash said:
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 "$".
 

FAQ: Inequality to represent minimum monthly income.

1. What is "inequality to represent minimum monthly income?"

Inequality to represent minimum monthly income is a mathematical concept that compares the minimum amount of money a person or household needs to earn in a month in order to meet their basic needs. It can also refer to the disparity between the minimum amount a person needs to earn and what they actually earn.

2. How is inequality to represent minimum monthly income measured?

This type of inequality is typically measured using a metric called the minimum income standard (MIS), which takes into account the cost of basic necessities such as housing, food, utilities, and healthcare. The MIS varies depending on factors such as location, family size, and household composition.

3. Why is it important to understand inequality to represent minimum monthly income?

Understanding this type of inequality can provide insight into the economic well-being and financial stability of individuals and households. It can also help identify areas where there may be a lack of access to basic necessities and inform policy decisions aimed at reducing poverty and promoting financial equality.

4. How does inequality to represent minimum monthly income impact society?

Inequality to represent minimum monthly income can have far-reaching effects on society. It can contribute to social and economic disparities, limit opportunities for upward mobility, and perpetuate cycles of poverty. It can also strain government resources and increase social and political tensions.

5. Can inequality to represent minimum monthly income be addressed or reduced?

Yes, there are various ways to address or reduce inequality to represent minimum monthly income. These can include policies such as increasing the minimum wage, providing access to affordable housing and healthcare, and implementing social safety nets for those in need. Additionally, promoting education and job training opportunities can help individuals and families increase their earning potential and improve their financial stability.

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