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

Analyze Given Options: We know that a^3 * b * c^2 < 0. Since a^3 and c^2 are both raised to an even power, they will always be non-negative (zero or positive) regardless of the sign of a and c. Therefore, the sign of the expression is determined by the sign of b and the sign of a (since a negative number raised to an odd power is negative). If b is positive, then a must be negative for the expression to be negative. If b is negative, then a can be either positive or negative. Option (A): Let's analyze the options given: (A) a 0: This could be true because if a is negative and raised to an odd power, it remains negative, and if b is positive, the expression would be negative. However, this does not cover all cases because b could also be negative.Option (D): (D) a 0 or a > 0 & b < 0: This must be correct because it covers all cases where the expression could be negative. If a is negative and b is positive, the expression is negative. If a is positive and b is negative, the expression is also negative. This option accounts for the fact that the sign of a^3 is determined by the sign of a, and the overall sign of the expression is determined by the sign of b.

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If a^3bc^2 < 0, which of the following must be correct? [Without calculator] $\newline$ (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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Math Problems

Algebra 1

Is (x, y) a solution to the system of linear inequalities?

Full solution

Q. If

a^3bc^2 < 0

, which of the following must be correct? [Without calculator]

\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 Given Options: We know that a^3 \cdot b \cdot c^2 < 0. Since $a^3$ and $c^2$ are both raised to an even power, they will always be non-negative (zero or positive) regardless of the sign of $a$ and $c$ . Therefore, the sign of the expression is determined by the sign of $b$ and the sign of $a$ (since a negative number raised to an odd power is negative). If $b$ is positive, then $a$ must be negative for the expression to be negative. If $b$ is negative, then $a$ can be either positive or negative.
Option (A): Let's analyze the options given: $\newline$ (A) a < 0 & c < 0: This is not necessarily true because $c$ could be positive and the expression could still be negative as long as $a$ is negative and $b$ is positive.
Option (B): (B) b < 0 & c < 0: This is not necessarily true because $c$ could be positive and the expression could still be negative as long as $a$ is negative and $b$ is positive.
Option (C): $C$ a < 0 & b > 0: This could be true because if $a$ is negative and raised to an odd power, it remains negative, and if $b$ is positive, the expression would be negative. However, this does not cover all cases because $b$ could also be negative.
Option (D): (D) a < 0 \& b > 0 or a > 0 \& b < 0: This must be correct because it covers all cases where the expression could be negative. If $a$ is negative and $b$ is positive, the expression is negative. If $a$ is positive and $b$ is negative, the expression is also negative. This option accounts for the fact that the sign of $a^3$ is determined by the sign of $a$ , and the overall sign of the expression is determined by the sign of $b$ .