In addition to using its own capital reserves for the development, Pinecrest Enterprises must have access to a line of credit. Robert, the manager of the local bank, is concerned about Pinecrest’s existing debt. The total debt is $72 million. Joe, Pinecrest’s CEO, believes the company can reduce the existing debt by $2.5 million per month. Robert indicates that the company can receive additional credit if its debt has been reduced to between $37 million and $52 million.
What is the range for the number n of monthly payments that correspond to this target? When writing your answer as an inequality in the form a less or equal than n less or equal than b, what are the values of a and b?
Compound inequality:
a<2.5n<2.5<52.
37<2.5n<52
Divide by 2.5:
14.8M<n<20.8M.
Correction:
a<2.5n<b.
37<2.5n<52
Divide by 2.5:
14.8M<n<20.8M.
What is the value of a & b? I don’t understand.
37<2.5n<52.
14.8<n<20.8.
So the compound inequality states that the number of monthly payments(n) is greater than 14.8 but less than 20.8.
a = 14.8(15).
b = 20.8(21).
To determine the range for the number of monthly payments that correspond to the target debt range, we can set up an inequality based on the given information.
Let's assume the number of monthly payments is represented by 'n'.
The initial debt is $72 million, and Joe believes the company can reduce the existing debt by $2.5 million per month. So, the equation for the remaining debt after 'n' months is:
Remaining Debt = Initial Debt - (Monthly Reduction × Number of Months)
Remaining Debt = 72 million - (2.5 million × n)
According to Robert, the company can receive additional credit if its debt has been reduced to between $37 million and $52 million. Therefore, we can set up the following inequality:
37 million ≤ Remaining Debt ≤ 52 million
Substituting the expression for the remaining debt from above:
37 million ≤ 72 million - (2.5 million × n) ≤ 52 million
Simplifying this inequality, we get:
-35 million ≤ -2.5 million × n ≤ -20 million
Now we can divide the entire inequality by -2.5 million to isolate 'n':
-35 million / -2.5 million ≥ n ≥ -20 million / -2.5 million
14 ≥ n ≥ 8
Therefore, the range of the number of monthly payments 'n' that correspond to the target debt range is 8 ≤ n ≤ 14.