Solve the inequality for x: (x – 4)(6 – x) ≥ 0

Solution

We are given the inequality: (x – 4)(6 – x) ≥ 0 Note that: (x – 4)(6 – x) = –(x – 4)(x – 6) So the inequality becomes: –(x – 4)(x – 6) ≥ 0 Multiply both sides by –1 (remember: inequality sign reverses): (x – 4)(x – 6) ≤ 0 We solve: (x – 4)(x – 6) ≤ 0 The critical points are: x = 4 and x = 6 Make a sign chart: ·        For x < 4: both factors negative → product positive ·        For 4 ≤ x ≤ 6: one factor negative, one positive → product negative or zero ·        For x > 6: both positive → product positive We want: (x – 4)(x – 6) ≤ 0 So, solution is: x ∈ [4, 6]