ЯсноПонятно24

To solve this inequality, we need to find the critical points where the expression is equal to zero and then determine the sign of the expression in each interval created by these critical points.

First, let's find the critical points by setting the expression equal to zero: x(6+x)(x-3/4) = 0

This expression will be equal to zero when x = 0, x = -6, and x = 3/4.

Now, let's test the expression in each interval created by these critical points:

1. Interval (-∞, -6): Choose x = -7: (-7)(-1)(-7-3/4) = (-7)(-1)(-31/4) = 217/4 > 0

2. Interval (-6, 0): Choose x = -1: (-1)(5)(-1-3/4) = (-1)(5)(-7/4) = 35/4 > 0

3. Interval (0, 3/4): Choose x = 1/2: (1/2)(6+1/2)(1/2-3/4) = (1/2)(13/2)(-1/4) = -13/16 < 0

4. Interval (3/4, ∞): Choose x = 1: (1)(7)(1-3/4) = (1)(7)(1/4) = 7/4 > 0

Therefore, the solution to the inequality x(6+x)(x-3/4) > 0 is x ∈ (-∞, -6) U (0, 3/4) U (3/4, ∞).