To solve the inequality x(3 - 4x) ≤ 0, we can follow these steps:
First, determine the critical points by setting each factor equal to zero and solving for x:
x = 0 (from x = 0)
3 - 4x = 0
Solving the second equation:
4x = 3
x = 3/4
So, the critical points are x = 0 and x = 3/4.
Next, create a number line and mark the critical points on it:
------------------o----o------------------
0 3/4
Now, choose a test point from each interval created by the critical points and evaluate the expression x(3 - 4x) for that test point:
For x < 0 (choose x = -1):
(-1)(3 - 4(-1)) = (-1)(3 + 4) = (-1)(7) = -7 < 0
For 0 < x < 3/4 (choose x = 1/2):
(1/2)(3 - 4(1/2)) = (1/2)(3 - 2) = (1/2)(1) = 1/2 > 0
For x > 3/4 (choose x = 1):
(1)(3 - 4(1)) = (1)(3 - 4) = (1)(-1) = -1 < 0
Determine the sign of x(3 - 4x) for each interval:
For x < 0, x(3 - 4x) < 0
For 0 < x < 3/4, x(3 - 4x) > 0
For x > 3/4, x(3 - 4x) < 0
Finally, write the solution to the inequality by considering the sign of x(3 - 4x) for each interval:
x(3 - 4x) ≤ 0 is true when x lies in the interval (0, 3/4] (including 0 and 3/4).
Therefore, the solution to the inequality x(3 - 4x) ≤ 0 is x ∈ [0, 3/4].