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Arithmetic operations include addition, subtraction, multiplication, and division. These are the fundamental processes used for calculations in mathematics and appear in nearly every math problem. Simplifying algebraic expressions often involves a combination of these operations.

For instance, in our exercise, we start with the expression \( -10-(-5)+7-2 \) which involves both addition and subtraction operations. An understanding of how to use these operations with whole numbers as well as negative numbers is pivotal. It's important to note that subtraction of a negative number, like \( -(-5) \), is equivalent to addition of the positive number, as two negatives make a positive. This is why \( -10-(-5) \) becomes \( -10+5 \).

This concept ensures that students recognize subtraction of a negative as the addition of its positive counterpart. Thus, rather than taking away, we are effectively adding, and this changes how we approach the problem.

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed in an algebraic expression.

In the absence of parentheses and exponents, as with the given exercise, we address the multiplication or division first (although there are none in this case), and then move on to addition and subtraction, performing them from left to right as they appear. The straightforward nature of our expressions \( -10+5+7-2 \) and \( -6+3+8-11 \) indicates that we simply solve them in the order we encounter them, respecting the operation signs.

By adhering to this order, we ensure that the arithmetic is performed accurately, allowing for clean, mistake-free simplification of expressions.

Negative numbers are numbers with a value less than zero, often representing a lack or shortage. They play a crucial role in algebraic expressions, especially when combined with positive numbers through various operations.

Understanding negative numbers is essential when simplifying algebraic expressions. For example, when we are subtracting a positive number from a negative number, such as \( -6-2 \), we are going further into the negatives, yielding \( -8 \). Alternatively, when adding a negative number to a positive number, like in \( 7-2 \), it’s critical to recognize that the overall value is decreasing, resulting in \( 5 \).

It’s also important to understand that adding two negative numbers together results in a more negative number. Meanwhile, the subtraction of a negative number, as earlier illustrated, is conceptually transforming the operation into the addition of a positive number. This nuance has a significant impact on simplifying expressions that contain negative numbers.