To solve the inequality (3x-6)(x-x^2) > 0, we first need to find the critical points where the expression equals 0.
Setting each factor equal to 0 gives us 3x-6 = 0 => 3x = 6 => x = x - x^2 = 0 => x(1-x) = 0 => x = 0, x = 1
Now we can plot these critical points on a number line:
----0---1---2---
Next, we need to test the intervals between these critical points to see where the expression is positive.
For x < 0: (3(-6))(((-6)-0)) = (-18)(-6) > For 0 < x < 1: (3(6))((6-(-6))) = (18)(12) > For 1 < x < 2: (3(6))((6-(-6))) = (18)(12) > For x > 2: (3(6))((6-0)) = (18)(6) > 0
Since the expression is greater than 0 for x < 0, 0 < x < 1, 1 < x < 2, and x > 2, the solution to the inequality is:
To solve the inequality (3x-6)(x-x^2) > 0, we first need to find the critical points where the expression equals 0.
Setting each factor equal to 0 gives us
3x-6 = 0 => 3x = 6 => x =
x - x^2 = 0 => x(1-x) = 0 => x = 0, x = 1
Now we can plot these critical points on a number line:
----0---1---2---
Next, we need to test the intervals between these critical points to see where the expression is positive.
For x < 0: (3(-6))(((-6)-0)) = (-18)(-6) >
For 0 < x < 1: (3(6))((6-(-6))) = (18)(12) >
For 1 < x < 2: (3(6))((6-(-6))) = (18)(12) >
For x > 2: (3(6))((6-0)) = (18)(6) > 0
Since the expression is greater than 0 for x < 0, 0 < x < 1, 1 < x < 2, and x > 2, the solution to the inequality is:
x < 0 or 0 < x < 1 or 1 < x < 2 or x > 2