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When we talk about solving absolute value equations, we're dealing with expressions that determine the distance of a number from zero on a number line. The absolute value is always non-negative, which means it can create two different scenarios when set equal to another expression. Let’s start with the equation \( |x-3|=|6-x| \). Process involves looking at the two possibilities: first, when the quantities inside the absolute values are equal \( x-3 = 6-x \), and second, when one is the negative of the other \( x-3 = -(6-x)\).

To find the solutions, we must tackle each possibility separately. The first scenario led to a valid solution where \( x \) was 4.5. However, the second case resulted in an algebraic contradiction, indicating that no valid solution comes from that route. Therefore, the sole solution is \( x = 4.5 \) which we found by utilizing basic algebra to isolate \( x \) on one side of the equation.

Moving on to equivalent expressions, it's essential to understand that these are different ways of writing the same mathematical relationship. In the context of absolute value equations, we often explore equivalent expressions by opening up the absolute value to consider both positive and negative scenarios.

In our example, dealing with these equivalent expressions meant setting \( x-3 \) equal to \( 6-x \) and its negative, \( -6+x \). By manipulating these expressions separately, we could find that they could be solutions to the original equation, as long as they maintain the balance of the equation after the absolute value signs are removed.

Algebraic contradictions occur when an operation on an equation produces a statement that is always false, such as \( 3=-6 \). When we stumble upon this during the solving process, it's a clear sign that the corresponding scenario doesn't yield a valid solution for the equation.

In the context of our absolute value equation, the second scenario led us straight to a contradiction, after simplifying down to an impossible equality. This contradiction is important—it tells us that the equation has no solution in that particular case and thus clarifies that not all conditions produce results.

The solution set of an equation contains all the values that satisfy it. For absolute value equations, the solution set could include two values, one for each scenario, or it might be a single value or even empty, depending on the scenarios after solving.

For our equation \( |x-3|=|6-x| \), the solution set is ultimately straightforward: \( \{4.5\} \). There's only one value because the second scenario doesn’t produce a solution, thanks to an algebraic contradiction. A key takeaway here is that the solution set for absolute value equations is not always intuitive. Careful analysis of both scenarios presented by the absolute value is crucial for the correct determination of the solution set.