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).
The maximum value of x + y is then given by the corners of the rectangle R, so the maximum is 3/11 + 15/11 = 18/11.