$2 x ^ { 3 } + 9 x ^ { 2 } - 27 x - 54 0$ A) $x \leq - 6 \text { or } - \frac { 3 } { 2 } \leq x \leq 3$ B) $x

Question

$2 x ^ { 3 } + 9 x ^ { 2 } - 27 x - 54 > 0$ A) $x \leq - 6 \text { or } - \frac { 3 } { 2 } \leq x \leq 3$ B) $x < - 6 \text { or } - \frac { 3 } { 2 } < x < 3$ C) $- 6 \leq x \leq - \frac { 3 } { 2 } \text { or } x \geq 3$ D) $- 6 < x < - \frac { 3 } { 2 } \text { or } x > 3$

Quizplus · Accepted Answer

To solve the inequality $2x^3 + 9x^2 - 27x - 54 > 0$, we factor it to find its roots and determine the intervals where the inequality holds true. Factoring gives us:$$2x^3 + 9x^2 - 27x - 54 = (x + 6)(x + \frac{3}{2})(x - 3) > 0$$The roots are $x = -6$, $x = -\frac{3}{2}$, and $x = 3$. Plotting these on a number line and testing intervals between and outside these roots, we find that the inequality is satisfied for $x < -6$, $-\frac{3}{2} < x < 3$, and $x > 3$. However, since the inequality is strictly greater than (>) and not greater than or equal to ($\geq$), the endpoints are not included, leading to the correct answer being $x < -6$ or $-\frac{3}{2} < x < 3$ or $x > 3$, which matches option D: $-6 < x < -\frac{3}{2}$ or $x > 3$.

2x3+9x2−27x−54>02 x ^ { 3 } + 9 x ^ { 2 } - 27 x - 54 > 02x3+9x2−27x−54>0

$2 x ^ { 3 } + 9 x ^ { 2 } - 27 x - 54 > 0$