I am doing an exercise for a Linear Programming Problem where a business is trying to maximise their profits based on the sale of armchairs and sofas.
In the question it is stated that market research has shown that between 4 to 9 armchairs should be held in stock for each 3 sofas held in stock.
$x=$armchairs and $y=$sofas
The armchair/sofa constraint given in the question is $4y\leq3x\leq9y$
Am I misunderstanding this completely or should the constraint not be $4x\leq3y\leq9x$?
I find an easy way to think of this is to plug in real numbers and see if it works. Say we have 3 sofas. Let's use your equation. 4x<3y<9x -> 4x<9<9x
This equation implies that we should have between 1 and 9/4ths of a armchair. What it SHOULD say is that we should have between 4 and 9. If we try the correct equation, we see that it does actually work.
4y<3x<9y -> 12<3x<27
4 < x < 9
This confusing need to switch variables happens a lot. Consider if I could buy 2 tacos for a dollar. At first glance, it may seem you can say
2*number of tacos = cost.
But plug in 1 taco, and you are paying 2 dollars. The equation actually reads "The cost equals twice the numbers of tacos bought." Moral of the story: check your equations carefully when interpreting!