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Understanding the order of operations is crucial when simplifying mathematical expressions, especially when they contain multiple steps and use a variety of arithmetic operations.

The standard order in which operations should be performed is denoted by the acronym PEMDAS:

• Parentheses first
• Exponents (i.e., powers and square roots, etc.)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This means that when faced with an expression such as
\[ [5-(-2-5)]-[2-(-1-4)] \]
you must first address the operations within the parentheses. This is why in the solution, the first step focuses on simplifying the expressions inside the brackets
\[ 5-(-2-5) \]and \[ 2-(-1-4) \].

By following the order of operations, you avoid common mistakes and ensure that you arrive at the correct answer, just as we reached \(1\) in the simplified expression.

The subtraction of integers often confuses students, particularly when negative signs are involved. The key to correctly subtracting integers is to remember that subtracting a negative number is the same as adding its opposite.

To illustrate, in the given problem, once we have simplified the expressions inside the brackets, we face a scenario where we must subtract a negative number:
\[ [-2]-[-3] \].

This can be rewritten as \( -2 + 3 \) because \( -(-3) = +3 \). This rule makes it easier to understand why we add the opposite number instead of subtracting a negative. Any time you see \( -- \) next to each other, you replace them with a \( + \) sign, which simplifies the process of finding the solution.

Simplifying algebraic expressions is a fundamental skill in algebra. An algebraic expression may include numbers, variables, and arithmetic operations.

In the context of our problem,

• The algebraic expressions given are
  \[ 5-(-2-5) \] and \[ 2-(-1-4) \].
• They are nested within brackets, which indicate they should be simplified as separate expressions before any other operation is performed on them.

The challenge often lies in correctly applying the rules for arithmetic operations, such as the subtraction of integers, as well as maintaining strict adherence to the order of operations. By attending to each of these aspects, as demonstrated in the step-by-step solution, we seamlessly reduced the complex-looking original expressions to the much simpler answer of \(1\).

It's essential to tackle algebraic expressions systematically, breaking them down according to the hierarchy of operations, to prevent errors and to improve your mathematical problem-solving skills.