Question 958510
A couple of those steps look strange, and your variable choice is confusing.


w and L for the dimensions.
{{{system(2L+2w=120, wL>=500)}}};
What can you do with that system?


{{{L+w=60}}}
{{{w=60-L}}}
Now, your shown steps make more sense.
Substitution:
{{{(60-L)L>=500}}}
{{{60L-L^2-500>=0}}}
{{{-60L+L^2+500<=0}}}
{{{L^2-60L+500<=0}}}, which comes from a parabola opening upward, the minimum below the horizontal axis, and you are interested in the part between the two real roots.


Discriminant, {{{3600-4*500=3600-2000=1600}}}.
{{{L=(60+- sqrt(1600))/2}}}
{{{L=(60+- 40)/2}}}
Taking as large as possible, this length can be between roots 10 and 50.


Interval Notation for your answer,
[10,50].