Question

a, b, c, d, and x are all non-zero integers. Is the product \( ax \cdot (bx)^2 \cdot (cx)^3 \cdot (dx)^4 \) negative? Statement (I): \( a < c < x < 0 \)
Statement (II): \( b < d < x < 0 \)

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

When evaluating the sign of a product with exponents, always simplify the expression first. Even powers of non-zero variables are always positive, allowing you to ignore them when determining the sign of the whole product.

Answer 1

The Correct Option is A

Concept: The sign of a product of multiple terms is determined by the parity of negative factors and the sign of each individual variable. Let the expression be \( P \). \[ P = ax \cdot (bx)^2 \cdot (cx)^3 \cdot (dx)^4 \]

Step 1: Simplify the expression \( P \). Expanding the powers: \[ P = (ax) \cdot (b^2 x^2) \cdot (c^3 x^3) \cdot (d^4 x^4) \] Grouping the variables: \[ P = (a \cdot b^2 \cdot c^3 \cdot d^4) \cdot x^{(1+2+3+4)} \] \[ P = (a \cdot b^2 \cdot c^3 \cdot d^4) \cdot x^{10} \]

Step 2: Determine the sign of the constant factors. Since \( b \) and \( d \) are non-zero integers, their even powers are always positive: \[ b^2 > 0, \quad d^4 > 0 \] Since \( x \) is a non-zero integer, \( x^{10} \) is always positive: \[ x^{10} > 0 \] Thus, the sign of \( P \) depends solely on the sign of \( a \cdot c^3 \): \[ \text{sign}(P) = \text{sign}(a \cdot c^3) \]

Step 3: Evaluate Statement (I): \( a < c < x < 0 \). Since \( a < 0 \) and \( c < 0 \), both are negative. Because \( c \) is negative, \( c^3 \) is negative: \[ a \cdot c^3 = (\text{negative}) \cdot (\text{negative}) = \text{positive} \] Since \( P \) is positive, we can definitively answer "No, the product is not negative." Thus, Statement (I) is sufficient.

Step 4: Evaluate Statement (II): \( b < d < x < 0 \). This statement provides information about \( b \) and \( d \), but provides no information regarding the signs or values of \( a \) or \( c \). Thus, Statement (II) is not sufficient. Statement (I) alone is sufficient.