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Prealgebra

Key Concepts

PrealgebraKey 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.

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
  • 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.
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