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3.7.2: Key Concepts

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    118905
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    Key Concepts

    3.1 Introduction to Integers

    • Opposite Notation
      • aa means the opposite of the number aa
      • The notation aa 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 negative sum positive
      When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

    3.3 Subtract Integers

    • Subtraction of Integers
      5353 –5(–3)–5(–3)
      22 –2–2
      2 positives 2 negatives
      When there would be enough counters of the color to take away, subtract.
      –53–53 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
    • Subtraction Property
      • ab=a+(−b)ab=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=bthenac=bc.ifa=bthenac=bc.
      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 c0c0
        If a=ba=b, then ac=bcac=bc.

    3.7.2: Key Concepts is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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