Final answer:
To solve the quadratic inequality x² + 6x ≥ 0, we factor it into x(x + 6) and find the critical points x = 0 and x = -6. Then, we test intervals and determine the solution is x ∈ (-∞, -6] U [0, ∞).
Step-by-step explanation:
To solve the quadratic inequality x² + 6x ≥ 0, we need to find the values of x where this inequality holds true. We factor the left-hand side of the inequality:
x(x + 6) ≥ 0
This factors into two parts, x and x + 6, which means the product is non-negative when both factors are either positive or negative. To determine where this occurs, we set each factor equal to zero and solve for x:
x = 0 and x + 6 = 0
Solving the second equation gives us x = -6. These are the critical points which divide the number line into intervals. We test each interval to see where the inequality is satisfied. We find that the inequality holds true for:
x ∈ (-∞, -6] U [0, ∞)