Problem 21 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

By carrying out an indirect proof and assuming that \(a \leq b\), a contradiction is identified when compared with the given information \(ac > bc\). Therefore, the assumption must be false and the original proposition that \(a > b\) must be true.

Step 1: Assume the Opposite

Begin by assuming the opposite of what you aim to prove. That means you should start with the assumption that \(a \leq b\). This is the contradiction to the proposition \(a > b\).

Step 2: Multiply Inequality by C

Now, multiply both sides of the inequality \(a \leq b\) with \(c\), knowing \(c > 0\). The inequality changes to \(ac \leq bc\).

Step 3: Identify Contradiction

By comparing the result in step 2 with the given condition \(ac > bc\), a contradiction is detected. It is not possible for \(ac\) to be both less than or equal to and greater than \(bc\) simultaneously.

Step 4: Conclude the Proof

After identifying a contradiction when assuming the opposite of the initial proposition, it can be concluded that the opposite must be false. Therefore, the statement \(a > b\) must be true under the given conditions.