Polynomial Inequality Equation

Question

1. x² + 4x + 4 ≥ 9
2. 2x³ + 13x² - 8x ≥ 52

Answer 1

x² + 4x + 4 ≥ 9

x² + 4x - 5 ≥ 0

(x + 5)(x - 1) ≥ 0

(x + 5)(x - 1) = 0 when x = 1 or -5

When x < -5, (x + 5)(x - 1) > 0

When -5 < x < 1, (x + 5)(x - 1) < 0

When x > 1, (x + 5)(x - 1) > 0

Solution set: (-∞, -5] ∪ [1, ∞)

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2x³ + 13x² - 8x - 52 ≥ 0

x²(2x + 13) - 4(2x + 13) ≥ 0

(2x + 13)(x² - 4) ≥ 0

(2x + 13)(x - 2)(x + 2) ≥ 0

(2x + 13)(x - 2)(x + 2) = 0 when x = -13/2, 2, and -2.

Check the sign of (2x + 13)(x - 2)(x + 2) to the left of -13/2, between -13/2 and -2, between -2 and 2, and to the right of 2.