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When faced with absolute value equations like \(|2x-3|=7\), it's essential to start by understanding the absolute value itself. The absolute value of a number is its distance from zero on the number line, regardless of direction, which means it's always non-negative. Solving an equation that includes an absolute value term requires us to consider the dual nature of absolute values: they encapsulate both the positive and negative scenarios that could produce the same absolute value.

Therefore, to solve these equations, we separate the equation into two different cases: one where the inside of the absolute value is positive, and one where it's negative. This is why in our Step 2, two separate equations were set up: one for each potential scenario. By handling these cases separately, you actually explore all possible solutions to the equation.

The first tactical step in dealing with an absolute value equation is to isolate the absolute value expression on one side of the equation. This means to get the absolute value all by itself, without any other terms or coefficients on its side of the equality. Clear isolation is crucial because it sets the stage for the next step, where we'll address the positive and negative cases. In the exercise \(|2x-3|=7\), the term \(2x-3\) is already set apart, which is perfect. Isolating the absolute value helps simplify the problem, as it allows us to focus on decoding what’s inside the absolute value without extraneous terms complicating matters.

Once the absolute value is isolated, the next step is to create two separate equations to handle the positive and negative cases, as absolute values deal with both. This involves setting the expression inside the absolute value equal to both the positive and negative value of the quantity on the other side of the equation.

For positive case: \(2x-3 = 7\)
For negative case: \(2x-3 = -7\)

These equations represent the two scenarios: where the expression inside the absolute value is either positive or negative but ends up having the same absolute value. By solving these cases, we cover all the ways that the variable \(x\) can interact with the absolute value to produce the given number, which in this exercise is 7.

The final and critical stage in solving absolute value equations is to verify that the potential solutions actually work in the original equation. Algebraic manipulation can sometimes introduce 'extraneous' solutions that don't hold true when substituted back into the original equation. To verify solutions, take each solution found from solving the positive and negative cases, and substitute them back into the original absolute value equation.

For instance, when substituting \(x = 5\) and \(x = -2\) into \(|2x-3|=7\), you'll find that both satisfy the equation, thus confirming their validity as solutions. If either substituted value did not fulfill the original equation, it would need to be discarded. Verification ensures that the solutions you present are legitimate within the context of the given problem.