Solving equations involving absolute values requires understanding how to handle the expressions inside the absolute value symbols. When dealing with a single absolute value equation, the goal is to isolate the absolute value expression on one side, resulting in a form like |x + 1| = 2. This allows the equation to be rewritten as two separate linear equations by setting the expression inside equal to both the positive and negative values of the number outside the absolute value. For example, x + 1 = 2 and x + 1 = -2 lead to solutions x = 1 and x = -3, respectively. These solutions are then combined into a solution set, denoted by curly brackets.

When an equation contains two absolute values set equal to each other, such as |x + 1| = |2x - 4|, a similar approach applies. The key is to recognize that if |A| = |B|, then either A = B or A = -B. This principle allows the original equation to be split into two separate equations without absolute values: one where the expressions inside the absolute values are equal, and another where one expression equals the negative of the other.

Applying this to the example, the first equation is x + 1 = 2x - 4, and the second is x + 1 = -(2x - 4). Solving the first equation involves isolating the variable by adding 4 to both sides, yielding x + 5 = 2x, then subtracting x from both sides to get 5 = x. The second equation requires distributing the negative sign: x + 1 = -2x + 4. Subtracting 1 from both sides gives x = -2x + 3, and adding 2x to both sides results in 3x = 3. Dividing both sides by 3 yields x = 1. Thus, the solution set for the original equation is {1, 5}.

This method highlights the importance of understanding absolute value properties and linear equation solving techniques. By transforming absolute value equations into equivalent linear forms, one can systematically find all possible solutions. This approach is essential for mastering absolute value equations, especially when multiple absolute values are involved, and reinforces foundational algebraic skills.

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