I'm confident the answer is x≥12, but is there more to the answer (like checking if there are any values for x), as it is a high marked question

Question

The specifications for a storage container state that the length is 1 metre more than triple the width and the height is 5 metres less than double the width. Find the range of possible dimensions for a volume of at least 8436 m3 using the algebraic method.

Answer 1

L=1+3W, H=2W-5

LWH > 8436

(1+3W)(W)(2W-5) = or > 8436

(3W^2 +W)(2W -5) = or > 8436

6W^3 -13W^2 -5W = or > 8436

6W^3 -13W^2 -5W - 8436 = or > 0

(W-12)(6W^2 +59W + 703) = or > 0

both factors must be > 0 or both < 0, or either = 0

W= or > 12 L = or > 37, H = or > 19

width is greater than or equal to 12 meters

length is greater than or equal to 37 meters

height is greater than or equal to 19 meters

or 0 < W < 12 and 6W^2 + 59W + 703 < 0 but that quadratic factor is never <0 for 0<W<12

the quadratic factor is never = 0 for any real value of W, since the discriminate <0