(х-2)(х+3)(х-1)≥0

To determine the intervals where the inequality (х-2)(х+3)(х-1)≥0 is satisfied, we can use the sign test method.

Step 1: Find the critical points by setting each factor equal to zero: x - 2 = 0 => x = 2 x + 3 = 0 => x = -3 x - 1 = 0 => x = 1

Step 2: Mark the critical points on a number line. -∞ -3 1 2 ∞

Step 3: Choose a test value from each interval and determine the sign of the function at that point. For the interval (-∞, -3): Choose x = -4 (-4 - 2)(-4 + 3)(-4 - 1) = (-6)(-1)(-5) = -30, which is negative.

For the interval (-3, 1): Choose x = 0 (0 - 2)(0 + 3)(0 - 1) = (-2)(3)(-1) = 6, which is positive.

For the interval (1, 2): Choose x = 1.5 (1.5 - 2)(1.5 + 3)(1.5 - 1) = (-0.5)(4.5)(0.5) = -0.56, which is negative.

For the interval (2, ∞): Choose x = 3 (3 - 2)(3 + 3)(3 - 1) = (1)(6)(2) = 12, which is positive.

Step 4: Determine the intervals where the function is either positive or zero. Based on the sign test results, the intervals where the inequality (х-2)(х+3)(х-1)≥0 is satisfied are (-3, 1) and (2, ∞).

Therefore, the solution to the inequality is x ∈ (-3, 1) U (2, ∞).