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An algebraic equation is a statement of equality between two algebraic expressions involving variables and constants. The primary objective when solving an algebraic equation is to find the value of the unknown variable that makes the equation true.

For instance, in the equation provided, we have an expression on each side of the equality involving the absolute values. The equation \( |x - 3| = |5 - x| \) is a classic example where the absolute value creates two distinct cases to consider. Solving for \(x\) in both scenarios requires isolating the variable on one side. This often involves performing similar operations on both sides of the equation to maintain the balance, which is a fundamental property of algebra.

Absolute value equations involve expressions within absolute value symbols, which denote the distance of a number from zero on the number line, regardless of the direction. To solve these equations, one must consider that the absolute value of a number is always nonnegative.

Here's a quick breakdown:

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• Isolate the absolute value expression, if necessary.
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• Write two separate equations: one for the positive case and another for the negative case.
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• Solve each equation independently as you normally would.
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• Check your solutions in the original equation, as not all solutions obtained from the negative case may be valid.

The equation \( |x - 3| = |5 - x| \) is split into \(x - 3 = 5 - x\) and \(x - 3 = - (5 - x)\) to accommodate for both the positive and negative cases of the absolute value. This approach is robust as it encompasses all possibilities for the equation's solution.

The solution set of an equation is the set of all values that satisfy the equation. When you solve an equation, you're essentially finding the solution set that makes the statement true. For some equations, the solution set may be a single number, an infinite set of numbers, or there may be no solution at all.

In the context of our example, we found \(x = 4\) from the first scenario, and no solution from the second scenario (since you cannot have \(0 = -2\)). Consequently, the solution set for the equation \( |x - 3| = |5 - x| \) contains only one element: \(\{4\}\). This illustrates that despite two potential cases from the absolute value expressions, only one yielded a valid solution. Always verify the solutions by plugging them back into the original equation to ensure they truly satisfy the equation.