A manufacturer wants to produce x balls and y boxes. Resource constrai

HOW TO ADD AND SUBTRACT INEQUALITIES
First and foremost, these are NOT equations, but inequalities. You cannot simply add and subtract inequalities in the same way as you can with equations. But let us see how we can...

This particular question asks for the maximum value of x+y.

1) We can ADD inequalities if the the signs face the same way:
\(7x+6y\leq{38000}\) and \(4x+5y\leq{28000}\)
...can be written as \((7x+6y)+(4x+5y)\leq{38000+28000}\)
...which simplifies to \(11x+11y\leq{76000}\)
\(x+y\leq{\approx{6900}}\)
Notice that the sign STAYS the same way.

2) We can SUBTRACT inequalities if the signs face the opposite way:
\(7x+6y\leq{38000}\) and \(28000\geq{4x+5y}\) (we switch the direction of the last inequality)
...can be written as \((7x+6y)-28000\leq{38000-(4x+5y)}\)
...which simplifies to \(7x+6y-28000\leq{38000-4x-5y}\) and \(11x+11y\leq{76000}\)
\(x+y\leq{\approx{6900}}\)
Notice that the we keep the first inequality's direction of the sign (the one we subtract FROM).

We get the same result either way. But of course, ADDING the inequalities makes the most sense for this particular question.