Question 415111: I can work no more than 36 hours per week at my two jobs. Housecleaning pays $6 per hour, and my sales job pays $12 per hour. I need to earn at least $302 per week to cover my expenses. I need to write a system of inequalities that shows the various numbers of hours I can work at each job.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x represent the number of hours housecleaning.
let y represent the number of hours selling.
your hours equation is x + y <= 36
your revenue equation is 6x + 12y >= 302
you have to satisfy both equations simultaneously.
Since you have 2 inequalities working against each other, the solution is very difficult to show, and you will have very numerous possibilities.
You pretty much have to look at the total hours you want to work and then solve the problem for those total hours.
You cannot work less than 25.16666667 hours, and all of those hours have to be sales hours.
That will give you a revenue of $302 per week.
It goes up from there.
For every 1 hour of sales that you don't do, you have to add 2 hours of housecleaning in order to maintain the minimum.
With 24.16666667 hours of sales, you will need 2 hours of housecleaning.
With 22.16666667 hours of sales, you will need 4 hours of housecleaning, etc.
To solve this problem in any meaningful way without going crazy, you have to pick the number of hours you want to work and then make your equation.
Example:
Your revenue equation will always be 6x + 12y >= 302
When x + y = 36, you can then substitute 36-x for y and solve the equation.
When x + y = 30, you can then substitute 30 - x for y and solve the equation.
Your break point will be when x + y = 25.16666667.
When that happens, you set y = 25.16666667 - x and you solve your revenue equation as follows:
6x + 12y <= 302
substitute 25.16666667 - x for y to get:
6x + 12 *(25.16666667 - x) >= 302
Simplify to get:
6x + 302 - 12x >= 302
combine like terms and subtract 302 from both sides of the equation to get:
-6x >= 0
multiply both sides of the equation by -1 to get:
6x <= 0
divide both sides of the equation by 6 to get:
x <= 0
since x can't be less than 0, the only solution is that x = 0.
So........
Your total hours have to be greater than or equal to 25.16666667 and less than or equal to 36.
Within that window, you have to pick the total hours you want to work and then solve the equation for that number of hours.
Assume you want to work 30 hours.
The equations are:
x + y = 30
This leads to y = 30-x
6x + 12y >= 302
Substitute 30-x for y to get:
6x + 12*(30-x) >= 302
solve the equation to get:
x <= 9.66666667
If you work 30 hours per week, x has to be less than or equal to 9.6666667 hours.
When you work 9.6666667 hours at x, then you will be working 20.33333333 hours at y and your revenue will be equal to $302.
If you work 10 hours at x, you will make 10*6 + 20*12 = $300 (not enough).
If your work 9 hours at x, you will make 9*6 + 21*12 = $306 (more than enough).
Bottom Line:
Total hours worked have to be greater than or equal to 25.16666667 and less than or equal to 36.
Within that window, pick the total hours you want to work and then the equations will tell you how many hours of each you have to work.
The equations you will work with are:
x + y = z, where 25.16666667 <= z <= 36
6x + 12y >= 302
Then solve for y = z - x and substitute in the second equation to solve for:
6x + 12*(z-x) >= 302
I don't know any simpler way to do this.
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