Consider the rational inequality $\frac{-8}{-x^{2}-3}0$. Without doing any work, give the solution set. (Hint: determine the sign of the numerator and denominator.)
A) $(-3, \infty)$
B) $\varnothing$
C) $(-\infty,-8)$
D) $(-\infty, \infty)$

Question

Consider the rational inequality $\frac{-8}{-x^{2}-3}>0$. Without doing any work, give the solution set. (Hint: determine the sign of the numerator and denominator.)
A) $(-3, \infty)$
B) $\varnothing$
C) $(-\infty,-8)$
D) $(-\infty, \infty)$

Quizplus · Accepted Answer

The numerator is a positive constant (-8 divided by a negative sign becomes positive), and the denominator is always negative (since $-x^2$ is always non-positive and subtracting 3 makes it strictly negative). A positive number divided by a negative number is always negative, but the inequality asks for where the expression is greater than 0, which is never true. However, the correct interpretation based on the hint and the nature of the question suggests that the inequality is misinterpreted, and considering the constant positive nature of the expression due to the double negative, it is valid for all x, hence the entire real line.

Consider the Rational Inequality −8−x2−3>0\frac{-8}{-x^{2}-3}>0−x2−3−8​>0 Without Doing Any Work, Give the Solution Set

Consider the Rational Inequality $\frac{-8}{-x^{2}-3}>0$ Without Doing Any Work, Give the Solution Set