Question

Use an indirect proof to prove that the conclusion is true. If \(a c>b c\) and \(c>0,\) then \(a>b\)

Short answer

By assuming the opposite of the statement \(a > b\) leads to a contradiction with the given conditions. Therefore, the statement 'If \(ac > bc\) and \(c > 0\), then \(a > b\)' is proven to be correct.

Step-by-step solution

Assume the Opposite

Start by assuming that the opposite of what you're trying to prove is true. In other words, assume that despite \(ac > bc\) and \(c > 0\), it is not the case that \(a > b\); instead \(a \leq b\).

Derive a Contradiction

Since \(c > 0\), you can multiply both sides of the inequality by \(c\) without reversing the direction of the inequality. If you do this, you get \(ac \leq bc\). But this contradicts the given assumption that \(ac > bc\).

Draw the Conclusion

Because assuming the opposite led you to a contradiction, you can conclude that the original statement must be true. Therefore, if \(ac > bc\) and \(c > 0\), then indeed \(a > b\).

Key concepts

Understanding Inequalities

Inequalities are mathematical expressions that demonstrate the relationship between two values when they are not equal. Unlike equations, which indicate that two expressions are equal, inequalities tell us that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. For example, the inequality expression \( a > b \) tells us that the value of \( a \) is greater than the value of \( b \).' Understanding how to manipulate inequalities is fundamental in solving various mathematical problems, particularly when dealing with real-world scenarios such as measurement, quantities, and finances.

When multiplying or dividing both sides of an inequality by a positive number, the direction of the inequality remains the same. However, if the number is negative, the inequality sign reverses. This principle is crucial in solving the given exercise.

Key Rules for Working with Inequalities:

• If \( a > b \) and \( c > 0 \), then \( ac > bc \).
• If \( a < b \) and \( c > 0 \), then \( ac < bc \).
• If you multiply or divide both sides by a negative number, reverse the inequality sign.

Proof by Contradiction

Proof by contradiction is a powerful tool in mathematical reasoning and problem-solving. It is based on the idea that if assuming the opposite of what you aim to prove leads to an impossibility or contradiction, then the original statement must be true. This method is particularly useful when a direct proof is challenging to find.

In the context of the exercise, the assumption that \( a \leq b \) led to \( ac \leq bc \) when multiplied by a positive number \( c \). This directly contradicted the initial condition \( ac > bc \) and thus proved the assumption false, confirming that in reality, \( a > b \).

Steps to Conduct a Proof by Contradiction:

• Clearly state the assumption which is the opposite of what needs to be proven.
• Using logical deductions and known theories, show that this assumption leads to a contradiction or an illogical conclusion.
• Since this assumption proves to be false, the original statement must therefore be true.

Employing Mathematical Reasoning

Mathematical reasoning goes beyond mere calculations and requires a logical thought process to deduce truths and establish the validity of statements. It is an essential skill in all mathematical proofs, including geometrical proofs, algebraic identities, and complex theorems.

The indirect proof from our exercise showcases mathematical reasoning by connecting a given set of premises (\( ac > bc \), \( c > 0 \) ) to arrive at a logical conclusion (\( a > b \)). Reasoning abilities come into play when we evaluate the implications of our assumptions and whether they align with established mathematical principles.

Key Aspects of Mathematical Reasoning:

• Understanding the properties and rules that govern numbers and operations.
• Identifying relevant information and disregarding what is not pertinent.
• Applying logic to deduce new information from what is already known.

Mathematical reasoning is a skill that is sharpened over time with practice and exposure to diverse problems. It is not only about finding the right answer but also about explaining why that answer makes sense within the realms of mathematical logic.