Question:
If ab>cd, and none of c, and d is equal to 0, which of the following must be true?
Indicate all such answers.
[Note: Select one or more answer choices]
If ab>cd, and none of c, and d is equal to 0, which of the following must be true?
Indicate all such answers.
[Note: Select one or more answer choices]
Indicate all such answers.
[Note: Select one or more answer choices]
Show Hint
When testing inequality statements, think about using a mix of positive and negative numbers for your counterexamples, as this is often where the statements fail. The most reliable statements are those that are direct algebraic manipulations based on the core rules of inequalities.
Updated On: Oct 3, 2025
- -ab
- |ab|>|cd|
- ba
- -ab<-cd
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
This question tests fundamental properties of inequalities, specifically how they behave under multiplication. We need to evaluate each statement to see if it logically follows from the given inequality ab > cd under all possible circumstances. Using counterexamples is an effective way to disprove statements that are not always true.
Step 2: Key Formula or Approach:
The core property of inequalities we will use is: If x > y, then for any negative number k, kx < ky. That is, multiplying or dividing an inequality by a negative number reverses the inequality sign. We will apply this rule and also test the other options with numerical examples to check their validity.
Step 3: Detailed Explanation:
We are given the inequality ab > cd. Let's analyze each option.
(A) -ab < cd: This is equivalent to ab > -cd. We are given ab > cd. Does ab > cd imply ab > -cd? Not always.
Counterexample: Let ab = 5 and cd = -10. Then 5 > -10 is true. The statement becomes -5 < -10, which is false. Therefore, A is not always true.
(B) |ab| > |cd|: This statement claims that the magnitude of ab is greater than the magnitude of cd. This is not always true.
Counterexample: Let ab = 5 and cd = -10. Then 5 > -10 is true. The statement becomes |5| > |-10|, which is 5 > 10, and this is false. Therefore, B is not always true.
(C) ba < dc: Since multiplication is commutative, ba = ab and dc = cd. So this statement is equivalent to ab < cd. This is the direct opposite of the given information (ab > cd). Therefore, C is never true.
(D) -ab < -cd: Let's start with the given inequality:
ab > cd
Multiply both sides by -1. According to the rules of inequalities, when we multiply by a negative number, we must reverse the direction of the inequality sign.
(-1) × (ab) < (-1) × (cd)
-ab < -cd
This statement is a direct consequence of the properties of inequalities and must be true. The conditions that c and d are non-zero are not needed for this derivation.
Step 4: Final Answer:
The only statement that must be true is (D).
This question tests fundamental properties of inequalities, specifically how they behave under multiplication. We need to evaluate each statement to see if it logically follows from the given inequality ab > cd under all possible circumstances. Using counterexamples is an effective way to disprove statements that are not always true.
Step 2: Key Formula or Approach:
The core property of inequalities we will use is: If x > y, then for any negative number k, kx < ky. That is, multiplying or dividing an inequality by a negative number reverses the inequality sign. We will apply this rule and also test the other options with numerical examples to check their validity.
Step 3: Detailed Explanation:
We are given the inequality ab > cd. Let's analyze each option.
(A) -ab < cd: This is equivalent to ab > -cd. We are given ab > cd. Does ab > cd imply ab > -cd? Not always.
Counterexample: Let ab = 5 and cd = -10. Then 5 > -10 is true. The statement becomes -5 < -10, which is false. Therefore, A is not always true.
(B) |ab| > |cd|: This statement claims that the magnitude of ab is greater than the magnitude of cd. This is not always true.
Counterexample: Let ab = 5 and cd = -10. Then 5 > -10 is true. The statement becomes |5| > |-10|, which is 5 > 10, and this is false. Therefore, B is not always true.
(C) ba < dc: Since multiplication is commutative, ba = ab and dc = cd. So this statement is equivalent to ab < cd. This is the direct opposite of the given information (ab > cd). Therefore, C is never true.
(D) -ab < -cd: Let's start with the given inequality:
ab > cd
Multiply both sides by -1. According to the rules of inequalities, when we multiply by a negative number, we must reverse the direction of the inequality sign.
(-1) × (ab) < (-1) × (cd)
-ab < -cd
This statement is a direct consequence of the properties of inequalities and must be true. The conditions that c and d are non-zero are not needed for this derivation.
Step 4: Final Answer:
The only statement that must be true is (D).
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