We want maximize the number of flowers in the garden (= maximize the sum of tulips and roses),
satisfying the imposed constraints.


Let's apply the most aggressive strategy.


Since the number of roses should be greater than or equal to the number of tulips,
we will take as much pairs (rose, tulip) as possible. Let the number of pairs be  p >= 0.

Each such pair requires 2+3 = 5 square feet of the garden area.

Then we will complement the pairs with separate roses. Let  r >= 0  be the numbers of separate roses.


Now the restriction regarding the area is

    5p + 3r <= 30  square feet.


The pairs will provide that the number of roses is not less than the number of tulips;
and each pair provides 2 flowers; so we are interested to maximize p and minimize  r >= 0.


Thus, in this inequality, we want to have integer r as small as possible, 
and due to this, to have p as great as possible.


Take r = 0.  Then  p = 30/5 = 6.


So, the optimal solution under given restriction is to plant 6 roses and 6 tulips.


Thus, the solution is 6 roses and 6 tulips, which gives the total of 6 + 6 = 12 flowers.


Notice that this solution satisfies all the constraints.