Answer
In the range of $(0,6]$ or $[19,25)$ feet.
Work Step by Step
Step 1. The perimeter of a rectangle is given by
$2(l+w)=50$, we have $l+w=25$
Step 2. Let $x=w$; we have $l=25-w=25-x$ with $0\lt x \lt 25$
Step 3. The area is given by $A=lw=x(25-x)=-x^2+25x$
Step 4. Let $A\leq114$; we have $-x^2+25x\leq114$ or $x^2-25x+114\geq0$
Step 5. Factoring the above inequality, we have $(x-6)(x-19)\geq0$ and the boundary points are $x=6,19$.
Step 6. Using the test points to examine signs of the left side across the boundary points, we have
$(0)...(+)...(6)...(-)...(19)...(+)...(25)$
Thus the solutions are $(0,6]$ or $[19,25)$ feet
Step 7. We conclude that the length of a side should be in the range of $(0,6]$ or $[19,25)$ feet.
