Q:

# x(3 β 4x) β€ 0

Accepted Solution

A:
To solve the inequality x(3 - 4x) β€ 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) β€ 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) β€ 0 is x β [0, 3/4].