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Understanding how to handle integers in mathematical operations is foundational in algebra. Integers include all whole numbers and their negative counterparts as well as zero. When adding or subtracting integers, there are specific rules that ensure you arrive at the correct answer.

For addition, if the signs are the same, you add the numerical values and keep the sign. Conversely, if the signs are different, you subtract the smaller numerical value from the larger one and take the sign of the number with the larger absolute value. Subtraction can be thought of as adding the opposite; for example, subtracting a positive is the same as adding a negative.

Working with Negative Numbers

Subtracting a negative number is akin to adding its positive counterpart. So, \( -a - (-b) = -a + b \). In the example \( -26 + 7 - 52 \), you combine the integers by first adding the positive number to the negative resulting in \( -19 \), and then subtract the remaining negative integer, \( -71 \).

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the exercise \( -26 + 7 - 52 \), there are no variables, but this operation follows the same principles that apply to expressions containing variables.

The ability to simplify expressions is crucial, as it forms the basis for solving more complex equations and understanding functions. While this example doesn't contain variables, it represents the kind of simplification you'd perform within an algebraic expression containing variables.

Importance of Simplification

Simplification generally involves combining like terms, which are terms with the same variable raised to the same power, and carrying out the specified operations. Always remember to do this step by step to avoid errors and to ensure the expression is reduced to its simplest form.

The order of operations is a rule that tells you the correct sequence to execute different operations in a mathematical expression. This rule 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).

In the given problem, there are only addition and subtraction operations. According to PEMDAS, these should be performed from left to right as they appear in the expression. If there had been multiple operations, applying the correct order of operations would prevent mistakes and give the correct result.

Applying PEMDAS

For instance, in a more complex expression like \( 3 + 2 \times (10 - 3)^2 \), you would first solve the operation inside the parentheses, then handle exponentiation, followed by multiplication, and finally addition. In the sample problem, \( -26 + 7 - 52 \), following the simple left to right rule for addition and subtraction led us to the correct answer of \( -71 \).