x(6+x)(x− 4 3 ​ )>0

To solve the inequality x(6+x)(x-4/3) > 0, we need to find the critical points where the expression equals 0 and then determine the intervals where the expression is positive.

First, find the critical points by setting each factor equal to 0: x = 0, x = -6, x = 4/3

Now, create a number line with these critical points: -6 | 0 | 4/3

Choose a test point in each interval and plug it into the expression to determine if it is positive or negative: For x < -6, choose x = -7: (-7)(-1)(-11/3) = 77/3 > 0, so this interval is positive. For -6 < x < 0, choose x = -1: (-1)(5)(-7/3) = 35/3 > 0, so this interval is positive. For 0 < x < 4/3, choose x = 1: (1)(7)(-1/3) = -7/3 < 0, so this interval is negative. For x > 4/3, choose x = 2: (2)(8)(2/3) = 32/3 > 0, so this interval is positive.

Therefore, the solution to the inequality x(6+x)(x-4/3) > 0 is x < -6 or -6 < x < 0 or x > 4/3.