If (x + 5)(x - 3) is negative, which of the following expressions must also be negative?

If (x + 5)(x - 3) is negative, which of the following expressions must also be negative?

• A. 3 - x
• B. 3x - 7
• C. (x + 7)(x - 3)
• D. (x + 5)(x - 5)

Answer: (A)

To understand why the selection of the expression "3 - x" is appropriate, we first need to analyze the conditions under which the product (x + 5)(x - 3) is negative. This occurs when the two factors have opposite signs, meaning one is positive and the other is negative. 1. **When is (x + 5)(x - 3) negative?** - The expression (x + 5) is equal to zero when x = -5, and (x - 3) is equal to zero when x = 3. - Therefore, the critical points are -5 and 3. - The intervals defined by these critical points are: (-∞, -5), (-5, 3), and (3, ∞). - By testing points in each interval: - In (-∞, -5), both (x + 5) and (x - 3) are negative, resulting in a positive product. - In (-5, 3), (x + 5) is positive, while (x - 3) is negative; hence, the product is negative. - In (3, ∞), both factors

To understand why the selection of the expression "3 - x" is appropriate, we first need to analyze the conditions under which the product (x + 5)(x - 3) is negative. This occurs when the two factors have opposite signs, meaning one is positive and the other is negative.

1. When is (x + 5)(x - 3) negative?

• The expression (x + 5) is equal to zero when x = -5, and (x - 3) is equal to zero when x = 3.

• Therefore, the critical points are -5 and 3.

• The intervals defined by these critical points are: (-∞, -5), (-5, 3), and (3, ∞).

• By testing points in each interval:

• In (-∞, -5), both (x + 5) and (x - 3) are negative, resulting in a positive product.

• In (-5, 3), (x + 5) is positive, while (x - 3) is negative; hence, the product is negative.

• In (3, ∞), both factors