For all x except x=5 the expression (x-5)^2 is strictly positive. The value x=5 is a solution since it sets the left hand side to zero.
Therefore, we can
(1) say that x=5 is one of the solutions of this inequality and
(2) for all other x we can reduce both sides of this inequality by positive (x-5)^2 without changing the comparison in the inequality getting a simpler inequality to solve:
(x-1)(x+3) <= 0
Next, we divide the set of real numbers into intervals by points that set each multiplier to zero (x=1 and x=-3).
Consider x as moving from -oo on the left to +oo on the right.
(a) While its less than -3, both x-1 and x+3 are negative and, therefore, their product is positive, not good for our inequality.
(b) As soon as x crosses -3, one of these multipliers, x+3, changes sign to positive, while the other, x-1, remains negative. Their product is negative, which is a solution.
(c) finally, as x crosses 1, both multipliers are positive, their product is positive too, not good for our inequality.
So, remaining points are [-3,1] and 5.
