Maximize and minimize : `Z = 15x + 30y` subject to the constraints : `x + y le 8, 2x + y ge 8, x - 2y ge 0, x, y ge 0`.

Maximize and minimize Z = x + 2y subject to the constraints x + 2y ge 100, 2x - y le 0, 2x + y le 200, x, y ge 0 .

Maximize and minimize Z = 3x + 2y - 3 subject to the constraints x + y ge 4, x + y le 12, x le 9, y le 9, x, y ge 0 .

Graphically minimize and maximize z = 3x + 4y subject to the constraints: x + y le 40 x + 2y le 80, x - 2y ge - 20 x, y ge 0 .

Maximize and minimize Z = 2x + 3y - 1 subject to the constraints x + y ge 5, x + y le 12, x le 10, y le 1 0, x, y ge 0 .

Graphically minimize and maximize z = 5x + 10y subject to the constraints: x +2y le 120, x+y le 60, x-2y ge 0 x, y ge 0 .

Maximize Z = 6x + 8y subject to the constraints x + y le 6, 3x + y ge 6, x - y ge 0, x, y ge 0 graphically.

Maximize Z = 22x + 44y subject to the constraints x + y ge 3, 3x + 8y le 24, x - y ge 0, x, y ge 0 .

Graphically minimize and maximize z = 7x + 5y subject to the constraints: x+y le 50, x + 2y le 100, x -y ge 10 x, y ge 0 .