This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — This question tests your understanding of how inequalities behave when dealing with negative numbers. The key condition is a < b < 0, which means both a and b are negative, and a is a smaller (more negative) number than b. For example, you can think of a = -3 and b = -2. Let's analyze each option: Given: a < b < 0 A. a^2 > ab Step 1: Consider the inequality a^2 > ab. We can subtract ab from both sides to get a^2 - ab > 0. Step 2: Factor out a from the expression: a(a - b) > 0. Step 3: Analyze the signs of the factors: Since a < 0, a is a negative number. Since a < b, if we subtract b from both sides, we get a - b < 0. So, (a - b) is also a negative number. Step 4: Multiply the signs: (negative number) × (negative number) = (positive number). So, a(a - b) is indeed positive, meaning a(a - b) > 0 is true. Therefore, a^2 > ab is true. Example: Let a = -3 and b = -2. a^2 = (-3)^2 = 9 ab = (-3)(-2) = 6 Is 9 > 6? Yes, it is. B. -(1)/(a) > -(1)/(b) Step 1: Start with the given a < b. Since a and b are both negative, taking the reciprocal of both sides reverses the inequality sign: (1)/(a) > (1)/(b) Step 2: Multiply both sides by -1. This also reverses the inequality sign: -(1)/(a) < -(1)/(b) This contradicts the statement in option B. Therefore, option B is false. Example: Let a = -3 and b = -2. -(1)/(a) = -(1)/(-3) = (1)/(3) -(1)/(b) = -(1)/(-2) = (1)/(2) Is (1)/(3) > (1)/(2)? No, (1)/(3) < (1)/(2). C. -(1)/(a) < (1)/(b) Step 1: Analyze the signs of each term. Since a < 0, (1)/(a) is negative. Therefore, -(1)/(a) is positive. Since b < 0, (1)/(b) is negative. Step 2: Compare a positive number to a negative number. A positive number is always greater than a negative number. So, -(1)/(a) > (1)/(b). This contradicts the statement in option C. Therefore, option C is false. Example: Let a = -3 and b = -2. -(1)/(a) = (1)/(3) (1)/(b) = -(1)/(2) Is (1)/(3) < -(1)/(2)? No, (1)/(3) > -(1)/(2). D. ab < b^2 Step 1: Consider the inequality ab < b^2. We can subtract b^2 from both sides to get ab - b^2 < 0. Step 2: Factor out b from the expression: b(a - b) < 0. Step 3: Analyze the signs of the factors: Since b < 0, b is a negative number. Since a < b, it means a - b is a negative number. Step 4: Multiply the signs: (negative number) × (negative number) = (positive number). So, b(a - b) is positive, meaning b(a - b) > 0. This contradicts the statement in option D. Therefore, option D is false. Example: Let a = -3 and b = -2. ab = (-3)(-2) = 6 b^2 = (-2)^2 = 4 Is 6 < 4? No, 6 > 4. Based on the analysis, only option A is true. The final answer is A What's next?