If a3bc2 0 (D) a 0 or a > 0 & b < 0"

Analyze inequality signs: Analyze the given inequality a^3 * b * c^2 < 0. Since a^3 is cubed, it will have the same sign as a. The term c^2 is squared, so it will always be non-negative (zero or positive). Therefore, the sign of the inequality is determined by the signs of a and b.Consider sign combinations: Consider the possible sign combinations for a and b that would make the inequality true. If a is positive, then b must be negative to make the product negative because c^2 is always non-negative. If a is negative, then b must be positive to make the product negative because c^2 is always non-negative.Match with options: Match the possible sign combinations to the given options. Option (A) states a 0, which is one possible correct combination. Option (D) states a 0 or a > 0 & b < 0, which includes both possible correct combinations.Determine correct answer: Determine the correct answer based on the analysis. Option (D) is the only option that includes all possible correct sign combinations for a and b that satisfy the inequality a^3 * b * c^2 < 0.

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If a^3bc^2 < 0, which of the following must be correct? $\newline$ (A) a < 0 \& c < 0 $\newline$ (B) b < 0 \& c < 0 $\newline$ (C) a < 0 \& b > 0 $\newline$ (D) a < 0 \& b > 0 or a > 0 \& b < 0

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Algebra 2

Solve quadratic inequalities

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Q. If

a^3bc^2 < 0

, which of the following must be correct?

\newline

(A)

a < 0 \& c < 0

\newline

(B)

b < 0 \& c < 0

\newline

(C)

a < 0 \& b > 0

\newline

(D)

a < 0 \& b > 0

a > 0 \& b < 0

Analyze inequality signs: Analyze the given inequality a^3 \cdot b \cdot c^2 < 0. Since $a^3$ is cubed, it will have the same sign as $a$ . The term $c^2$ is squared, so it will always be non-negative (zero or positive). Therefore, the sign of the inequality is determined by the signs of $a$ and $b$ .
Consider sign combinations: Consider the possible sign combinations for $a$ and $b$ that would make the inequality true. $\newline$ If $a$ is positive, then $b$ must be negative to make the product negative because $c^2$ is always non-negative. $\newline$ If $a$ is negative, then $b$ must be positive to make the product negative because $c^2$ is always non-negative.
Match with options: Match the possible sign combinations to the given options. $\newline$ Option (A) states a < 0 and c < 0, but $c$ 's sign does not matter since $c^2$ is always non-negative. $\newline$ Option (B) states b < 0 and c < 0, but this does not consider the sign of $a$ . $\newline$ Option (C) states a < 0 and b > 0, which is one possible correct combination. $\newline$ Option (D) states a < 0 \& b > 0 or c < 0 $0$ , which includes both possible correct combinations.
Determine correct answer: Determine the correct answer based on the analysis. $\newline$ Option (D) is the only option that includes all possible correct sign combinations for $a$ and $b$ that satisfy the inequality a^3 \cdot b \cdot c^2 < 0.