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Key Concepts

Key Concepts

3.1 Introduction to Integers

  • Opposite Notation
    • −a−a means the opposite of the number aa
    • The notation −a−a is read the opposite of a.a.
  • Absolute Value Notation
    • The absolute value of a number nn is written as |n||n|.
    • |n|≥0|n|≥0 for all numbers.

3.2 Add Integers

  • Addition of Positive and Negative Integers
    5+35+3 −5+(−3)−5+(−3)
    both positive, sum positive both negative, sum negative
    When the signs are the same, the counters would be all the same color, so add them.
    −5+3−5+3 5+(−3)5+(−3)
    different signs, more negatives different signs, more positives
    Sum negativesum positive
    When the signs are different, some counters would make neutral pairs; subtract to see how many are left.
    This table demonstrates the rules for adding integers, illustrating examples and results for sums involving numbers with both same and different signs.

3.3 Subtract Integers

  • Subtraction of Integers
    5–35–3 –5–(–3)–5–(–3)
    22 –2–2
    2 positives 2 negatives
    When there would be enough counters of the color to take away, subtract.
    –5–3–5–3 5–(–3)5–(–3)
    –8–8 88
    5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives
    need neutral pairs need neutral pairs
    When there would not be enough of the counters to take away, add neutral pairs.
    Table 3.13 This table illustrates integer subtraction methods, covering direct subtraction and cases requiring neutral pairs, with examples and conceptual steps.
  • Subtraction Property
    • a−b=a+(−b)a−b=a+(−b)
    • a−(−b)=a+ba−(−b)=a+b
  • Solve Application Problems
    • Step 1. Identify what you are asked to find.
    • Step 2. Write a phrase that gives the information to find it.
    • Step 3. Translate the phrase to an expression.
    • Step 4. Simplify the expression.
    • Step 5. Answer the question with a complete sentence.

3.4 Multiply and Divide Integers

  • Multiplication of Signed Numbers
    • To determine the sign of the product of two signed numbers:
      Same Signs Product
      Two positives
      Two negatives      		 Positive
      Positive

      Different Signs Product
      Positive • negative
      Negative • positive      		 Negative
      Negative
  • Division of Signed Numbers
    • To determine the sign of the quotient of two signed numbers:
      Same Signs Quotient
      Two positives
      Two negatives      		 Positive
      Positive

      Different Signs Quotient
      Positive • negative
      Negative • Positive      		 Negative
      Negative
  • Multiplication by −1−1
    • Multiplying a number by −1−1 gives its opposite: −1a=−a−1a=−a
  • Division by −1−1
    • Dividing a number by −1−1 gives its opposite: a÷(−1)=−aa÷(−1)=−a

3.5 Solve Equations Using Integers; The Division Property of Equality

  • How to determine whether a number is a solution to an equation.
    • Step 1. Substitute the number for the variable in the equation.
    • Step 2. Simplify the expressions on both sides of the equation.
    • Step 3. Determine whether the resulting equation is true.
      • If it is true, the number is a solution.
      • If it is not true, the number is not a solution.
  • Properties of Equalities
    Subtraction Property of Equality Addition Property of Equality
    For any numbersa,b,c,For any numbersa,b,c,
    ifa=bthena−c=b−c.ifa=bthena−c=b−c.    		 For any numbersa,b,c,For any numbersa,b,c,
    ifa=bthena+c=b+c.ifa=bthena+c=b+c.
  • Division Property of Equality
    • For any numbers a,b,c,a,b,c, and c≠0c≠0
       If a=ba=b, then ac=bcac=bc.