Question

if $x^2 - 6x - 27 > 0$, which statement is correct? a $-9 < x < 3$ b $-3 < x < 9$ c $-1 < x < 9$ d $x < -9$ or $x > 3$ e $x < -3$ or $x > 9$

Explanation:

Step1: Factor the quadratic expression

$x^2 - 6x - 27 = (x + 3)(x - 9)$

Step2: Find critical points

Set $(x + 3)(x - 9) = 0$, so $x = -3$ or $x = 9$

Step3: Test intervals for inequality

• For $x < -3$: $(x+3)<0, (x-9)<0$, product $>0$ (satisfies)
• For $-3 < x < 9$: $(x+3)>0, (x-9)<0$, product $<0$ (fails)
• For $x > 9$: $(x+3)>0, (x-9)>0$, product $>0$ (satisfies)

Answer:

E. $x < -3$ or $x > 9$