(a) Let \(R\) be the region consisting of the points \((x, y) \) that satisfy \(6 \le 3x - 5y \le 6 \) and \(-3 \le x + 2y \le 3\). Plot the region \(R\), and identify the vertices of polygon \(R\).
(b) Find the maximum value of \(x+y\) among all points \((x, y) \) in \(R\).
First, we can solve the inequalities as if they were equalities, so we solve the equation 3x - 5y = -6 and x + 2y = 3. Eliminating y, we get x = 3/11. Then y = 15/11, so (x,y) = (3/11, 15/11) is one corner of the region R. The inequalities are symmetric around the origin, so the region R is also symmetric around the region. So R is the rectangle with vertices (-3/11, 15/11), (3/11, 15/11), (-3/11, 15/11), and (-3/11, -15/11).