Solving problems involving integers requires understanding and applying arithmetic operations (addition, subtraction, multiplication, and division), along with following specific rules for dealing with positive and negative numbers. Here’s a step-by-step guide:

Steps to Solve Integer Problems

1. Understand the Problem: Identify the operations (addition, subtraction, etc.) and determine whether the integers are positive or negative.

2. Follow the Rules of Signs:

   • Addition:

     • Same signs: Add the absolute values and keep the common sign.
       Example: $-3 + (-5) = -8$.
     • Different signs: Subtract the smaller absolute value from the larger absolute value and take the sign of the larger absolute value. Example: $-3 + 5 = 2$.

   • Subtraction:

     • Change the subtraction to addition by reversing the sign of the second integer, then follow the addition rules. Example: $5 - (-3) = 5 + 3 = 8$.

   • Multiplication:

     • Same signs: The product is positive. Example: $(-3) \times (-5) = 15$.
     • Different signs: The product is negative. Example: $(-3) \times 5 = -15$.

   • Division:

     • Same signs: The quotient is positive. Example: $(-12) \div (-4) = 3$.
     • Different signs: The quotient is negative. Example: $12 \div (-4) = -3$.

3. Follow Order of Operations: Use the PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) rule for solving equations involving integers.

4. Check for Special Cases:

   • Zero as a factor in multiplication always results in 0.
   • Division by zero is undefined.

5. Simplify Your Answer: Always express the final answer in the simplest form.

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Example Problem

Problem: Solve $-7 + 3 \times (-2) - (-4)$.
Solution:

1. Apply multiplication: $3 \times (-2) = -6$.
2. Rewrite: $-7 + (-6) - (-4)$.
3. Simplify subtraction: $-7 - 6 + 4$.
4. Combine step by step:
   • $-7 - 6 = -13$,
   • $-13 + 4 = -9$.

Answer: $-9$.

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Questions to Explore

1. What are the common mistakes in handling signs for addition and subtraction of integers?
2. How can integer properties (associative, commutative, and distributive) simplify complex problems?
3. Why does multiplication or division of two negative integers result in a positive value?
4. How does the order of operations influence the outcome of integer problems?
5. What are some practical examples of integer problems in real-life scenarios?

Tip: Always write down intermediate steps to reduce errors when solving problems with multiple integers.

Would you like more detailed examples or have any specific integer-related questions?