There are three zeros of the inequality: 3, 5, and 0.
None of these will be solutions to the inequality, because there is no equal sign in the problem,
however, there are four partitions of the real number line:
x < 0 0 < x < 3 3 < x < 5 x > 5
Try a number smaller than 0, say -1, and place this number into the inequality:
(x - 3)(x - 5)x < 0 ---> (-1 - 3)(-1 - 5)(-1) < 0 ---> (-4)(-6)(-1) < 0 ---> -24 < 0
This is true!, so the whole region x < 0 is part of the solution set.
Next, try a number in the region 0 < x < 3, say 1:
(x - 3)(x - 5)x < 0 ---> (1 - 3)(1 - 5)(1) < 0 ---> (-2)(-4)(1) < 0 ---> 8 < 0
This is not true!, so this whole region is not part of the solution set.
Now, try a number in the region 3 < x < 5, say 4:
(x - 3)(x - 5)x < 0 ---> (4 - 3)(4 - 5)(4) < 0 ---> (1)(-1)(4) < 0 ---> -4 < 0
This is true!, so the whole region 3 < x < 5 is part of the solution set.
Next, try a number in the region x > 5, say 8:
(x - 3)(x - 5)x < 0 ---> (8 - 3)(8 - 5)(8) < 0 ---> (5)(3)(8) < 0 ---> 120 < 0
This is not true!, so this whole region is not part of the solution set.
The solution set consists of these two regions: x < 0 and 3 < x < 5.
