#1**0 **

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.

geno3141 Jun 23, 2020