Evaluating Complex Integer Expressions

This problem asks us to find the value of a mathematical expression involving the addition and subtraction of several integers, including negative numbers.

The given expression is:

\(-261 + (-380) – (-521) + 821 – (-121)\)

To evaluate this expression, we need to simplify it step-by-step by following the rules of integer operations.

Step-by-Step Evaluation of the Integer Expression

1. Remove Parentheses: We start by removing the parentheses. Remember that adding a negative number is the same as subtracting, and subtracting a negative number is the same as adding.
   • \(+(-380)\) becomes \(-380\)
   • \( – (-521)\) becomes \(+521\)
   • \( – (-121)\) becomes \(+121\)
   So the expression becomes: \(-261 - 380 + 521 + 821 + 121\)
2. Group Positive and Negative Numbers: It is often helpful to group the positive numbers together and the negative numbers together.
   • Negative numbers: \(-261, -380\)
   • Positive numbers: \(+521, +821, +121\)
3. Sum the Negative Numbers: Add the absolute values of the negative numbers and keep the negative sign. \(-261 - 380 = -(261 + 380) = -641\)
4. Sum the Positive Numbers: Add the positive numbers together. \(521 + 821 + 121\) \((521 + 821) + 121 = 1342 + 121 = 1463\)
5. Combine the Sums: Now, we combine the sum of the negative numbers and the sum of the positive numbers. \(-641 + 1463\)
6. Perform the Final Subtraction: Since the signs are different, we subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. Here, \(|1463| > |-641|\), so the result will be positive. \(1463 - 641\)

Let's perform the subtraction:

The result of the subtraction is \(822\).

Therefore, the value of the expression \(-261 + (-380) – (-521) + 821 – (-121)\) is \(822\).

Let's review the calculation steps:

\(-261 - 380 + 521 + 821 + 121\)

Combine negatives: \(-261 - 380 = -641\)

Combine positives: \(521 + 821 + 121 = 1463\)

Final result: \(-641 + 1463 = 1463 - 641 = 822\)

Revision Table: Key Rules for Integer Arithmetic

Additional Information: Properties of Integer Operations

Understanding basic properties helps simplify expressions:

• Commutative Property of Addition: The order in which integers are added does not change the sum. \(a + b = b + a\). Example: \(-5 + 3 = 3 + (-5) = -2\).
• Associative Property of Addition: The way integers are grouped when adding does not change the sum. \((a + b) + c = a + (b + c)\). Example: \((\(-2 + 3) + 4 = -2 + (3 + 4)\), so \(1 + 4 = -2 + 7 = 5\).
• Identity Property of Addition: The sum of any integer and zero is the integer itself. \(a + 0 = 0 + a = a\). Example: \(-7 + 0 = -7\).

These properties allow us to rearrange and group the terms in our expression \( -261 - 380 + 521 + 821 + 121 \) as \((-261 - 380) + (521 + 821 + 121)\), which simplifies the calculation as shown in the steps above.