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The goal of solving an absolute value equation, like \(|x-1| = 2\), is to find all possible values of \(x\) that make the equation true. Absolute value describes a number's distance from zero on the number line, regardless of direction. In basic terms, the expression within the absolute value signs, \(x-1\), can either be equal to the positive distance 2 or the negative distance -2 from zero. Hence, for this specific equation, we consider two separate cases:

• One where \(x-1 = 2\)
• Another where \(x-1 = -2\)

These cases represent all possible ways to get an absolute value of 2 from a number, leading us to pinpoint the precise values of \(x\) by solving each one. This approach ensures that no potential solution for \(x\) is overlooked.

Understanding how distance on a number line relates to absolute value is crucial in grasping absolute value equations. The absolute value of a number indicates how far the number is from zero, irrespective of direction. For \(|x-1| = 2\), this means either a shift 2 units to the right or 2 units to the left from 1.

• If we move right, we land at 3, making \(x-1 = 2\) or \(x = 3\).
• If we shift left, we find ourselves at -1, making \(x-1 = -2\) or \(x = -1\).

Both these conditions return the absolute value of 2 due to their symmetric positions around the pivot point of 1, corresponding to the nature of absolute value equations.

When tackling solution cases for equations in absolute value contexts, it is important to set up equations based on each scenario the absolute expression results in. For the equation \(|x-1| = 2\), our two cases are:

• Case 1: \(x-1 = 2\)
• Case 2: \(x-1 = -2\)

Let's solve these step-by-step. First, in Case 1, we solve \(x-1 = 2\) by adding 1 to both sides, giving us \(x = 3\). Next, in Case 2, solving \(x-1 = -2\) also involves adding 1 to both sides, resulting in \(x = -1\).
Combining these solutions, \(x = 3\) and \(x = -1\) are the numbers that satisfy the initial absolute value equation. This dual approach ensures checking all mathematical possibilities delivered by the absolute value.