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

Given Expression Analysis: We are given that a^3 * b * c^2 < 0. Since a^3 and c^2 are both raised to even powers, 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). Sign Determination: If a is positive, then a^3 is positive. If a is negative, then a^3 is negative. Since c^2 is always non-negative, it does not affect the sign of the expression. Therefore, the sign of the expression is negative only if either a is negative or b is negative, but not both.Option Analysis: We can now analyze the options given: (A) If a 0. This could make the expression negative, but it does not consider the sign of b. (B) If b 0, then a^3 0. This makes the expression negative, which is a possibility. (D) If a 0, or a > 0 and b 0, or a^3 > 0 and b < 0. Both cases result in a negative expression, which is a possibility.Final Conclusion: Since the expression a^3 * b * c^2 < 0 must be negative, and the sign of c^2 does not influence the sign of the expression, we can conclude that either a must be negative and b positive, or a positive and b negative. Therefore, option (D) is the correct answer because it is the only option that covers both possible scenarios where the expression is negative.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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

View step-by-step help

Home

Math Problems

Algebra 2

Solve quadratic inequalities

Full solution

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

Given Expression Analysis: We are given that a^3 \cdot b \cdot c^2 < 0. Since $a^3$ and $c^2$ are both raised to even powers, 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).
Sign Determination: If $a$ is positive, then $a^3$ is positive. If $a$ is negative, then $a^3$ is negative. Since $c^2$ is always non-negative, it does not affect the sign of the expression. Therefore, the sign of the expression is negative only if either $a$ is negative or $b$ is negative, but not both.
Option Analysis: We can now analyze the options given: $\newline$ (A) If a < 0 and c < 0, then a^3 < 0 and c^2 > 0. This could make the expression negative, but it does not consider the sign of $b$ . $\newline$ (B) If b < 0 and c < 0, then b < 0 makes the expression negative, which is a possibility, but c < 0 is irrelevant because $c^2$ is always non-negative. $\newline$ (C) If a < 0 and c < 0 $1$ , then a^3 < 0 and c < 0 $1$ . This makes the expression negative, which is a possibility. $\newline$ (D) If a < 0 and c < 0 $1$ , or c < 0 $6$ and b < 0, then either a^3 < 0 and c < 0 $1$ , or a^3 < 0 $0$ and b < 0. Both cases result in a negative expression, which is a possibility.
Final Conclusion: Since the expression a^3 \cdot b \cdot c^2 < 0 must be negative, and the sign of $c^2$ does not influence the sign of the expression, we can conclude that either $a$ must be negative and $b$ positive, or $a$ positive and $b$ negative. Therefore, option $(D)$ is the correct answer because it is the only option that covers both possible scenarios where the expression is negative.