## Systems of linear inequalities

A friend does $$15 an hour on his first job and$$11 per hour on his second job. His goal is to earn at least \$ 600 a week. He does not want to work less than 55 hours in a week.

## Inequalities – Why is Reduce in Mathematica the solution in other symbols that are not present in the original equation?

Here is an example:

``````To reduce[P/Q[P/Q[P/Q[P/Q< (P - X)/(Q - X) && X > 0 && P> 0 && Q> 0 && P> Q, {X}, integers]
``````

The solution to the above equation is:

``````(C[1] | C[2] | C[3])[Element] Integers &&
C[1] > = 0 &&[2] > = 0 &&
C[3] > = 0 && P = 3 + C[1] + C[2] + C[3] &&
Q == 2 + C[1] + C[2] &&
X = 1 + C[2]
``````

How do I interpret it? What is `C` and why are the numbers 1, 2 and 3 in brackets? The Reduction documentation does not use this notation in any of the examples. https://reference.wolfram.com/language/ref/Reduce.html

## Inequalities – Find the maximum of \$ f (x) \$

The following problem is due to a problem that I encountered while studying inequality. I find it hard to prove

To let $$n$$ give positive integer,$$(n ge 2)$$,and
$$f (x) = (nx) ln {(n + x + 1)} – (nx) ln {(nx)}, 0 le x le n-1, x in N ^ {+$$
Find the maximum of $$f (x)$$,from where $$x$$ be positive integers.

since
$$f & # 39; (x) = ln { left ( dfrac {n-x} {n + x + 1} right)} + dfrac {2n + 1} {n + x + 1}$$

I suspect when $$x = lfloor dfrac {n + 1} {2} rfloor$$ is maximum

## Inequalities – Find the maximum of \$ S = sum_ {1 leq i

The spot $$P_1, P_2, cdots, P_ {2018}$$ is placed inside or at the boundary of a certain regular pentagon. Find all placement methods such that $$S = sum_ {1 leqi takes the maximum value

His question can be a famous problem