Determine the solution for x²-3x-28≥0

(- ∞, - 4 ] ∪ [7, + ∞ )

Step-by-step explanation:

Determine the zeros by equating to zero, that is

x² - 3x - 28 = 0 ← in standard form

(x - 7)(x + 4) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 7 = 0 ⇒ x = 7

x + 4 = 0 ⇒ x = - 4

The zeros divide the domain into 3 intervals

(- ∞ , - 4 ], [ - 4, 7 ], [ 7, + ∞)

Choose a test point in each interval and compare value with required solution

x = - 5 : (- 5)² - 3(- 5) - 28 = 25 + 15 - 28 = 12 > 0

x = 0 : 0 - 0 - 28 = - 28 < 0

x = 10 : 10² - 3(10) - 28 = 100 - 30 - 28 = 42 > 0

Solutions are the first and third interval, that is

x ∈ (- ∞, - 4 ] ∪ [ - 7, + ∞)