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Simplifying expressions is a foundational skill in algebra. To simplify, you need to look inside the brackets and work with the numbers and operators inside them.
In our example, we have two expressions in brackets: \([-1 - (-3)]\) and \([-1 + (-3)]\).
The first step is to handle the subtraction of a negative number. When you see \(-(-3)\), it becomes \(+3\) because subtracting a negative is like adding a positive. So, \(-1 - (-3)\) simplifies to \(-1 + 3\), which equals \(2\).
For \([-1 + (-3)]\), you'll combine the numbers following the addition and subtraction rules. Adding a negative number is the same as subtracting the absolute value, hence \(-1 + (-3) = -1 - 3 = -4\).

Breaking down expressions into simpler components makes them easier to manage and sets the stage for further operations.
The aim is to have simpler, cleaner values that can be easily worked with in subsequent calculations.

Once expressions inside brackets are simplified, you often multiply the results. Multiplying integers follows straightforward rules.
You have positive and negative integers, and their combination affects your results. Here's a quick reminder:

• Positive times positive equals positive. \(3 \times 2 = 6\)
• Negative times positive equals negative. \(-3 \times 2 = -6\)
• Negative times negative equals positive. \(-3 \times -2 = 6\)

In the example, after simplifying the expressions to \(2\) and \(-4\), multiplying these gives \(2 \times -4 = -8\).
The positive multiplied by a negative results in a negative. And we can remember this through concepts like these:

• You are taking "two groups" of \(-4\), which results in \(-8\).
• Opposite signs produce negative outcomes.

Understanding how to manage signs is crucial, as it greatly impacts the results in mathematical operations.

Brackets in algebra indicate that the expression within them should be treated as a single unit. They are essential for order and clarity in mathematical expressions.
Using brackets helps dictate the order of operations and ensures calculations are executed correctly. In algebra, operations within brackets should be completed before dealing with those outside.

In our expression \([-1 - (-3)][-1 + (-3)]\), we first simplified the expressions inside the brackets. This is a direct application of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Employing brackets creates a structured approach and prevents errors in calculations. By taking it step by step, you focus on narrower parts of the problem, making it less overwhelming and easier to solve.
Recognizing their role can greatly improve problem-solving efficiency in algebra.