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Basic algebra is a fundamental part of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and express general relationships. For example, algebra allows us to represent an unknown value as a variable (typically an x or y) and find that value through the process of simplification and solving.

In the given exercise, we have the expression \[2+(-3)][-1-(-3)]\], where numbers and operations are already specified. In algebraic expressions like this, we often deal with a combination of integers, both positive and negative, and various operations such as addition, subtraction, and multiplication. Simplifying such expressions requires an understanding of how to combine these elements skillfully, which leads us to the key concepts of the order of operations and working with negative numbers.

The order of operations is critical in simplifying algebraic expressions correctly. It is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule helps us understand which parts of an expression to simplify first.

When we look at our example, \[2+(-3)][-1-(-3)]\], we first address the expressions inside the brackets (Parentheses). Once we have simplified the expressions within each set of parentheses, we proceed to the next operation, in this case multiplication of the resolved terms from each bracket. It's essential to follow this order to arrive at the correct solution.

Understanding Negative Numbers

Negative numbers are an integral part of algebra and can be a source of confusion if not handled properly. A negative number is a number that is less than zero, represented with a minus sign. When you add a negative number, it's equivalent to subtraction, and subtracting a negative number turns into addition, which can be counterintuitive at first.

In the example \[2+(-3)][-1-(-3)]\], we must correctly interpret the negative signs.

• For \(2+(-3))\), we're effectively subtracting 3 from 2.
• In \(\-1-(-3))\), subtracting negative three is the same as adding three to negative one.

This interpretation is pivotal in simplifying expressions with negative numbers and leads to the correct solution when combined with the order of operations.