3.7.2: Key Concepts

Key Concepts

3.1 Introduction to Integers

• Opposite Notation
  • $-a$ means the opposite of the number $a$
  • The notation $-a$ is read the opposite of $a.$
• Absolute Value Notation
  • The absolute value of a number $n$ is written as $\left|n\right|$.
  • $\left|n\right|\ge 0$ for all numbers.

3.2 Add Integers

• Addition of Positive and Negative Integers

  $5+3$	$\mathrm{-5}+\left(\mathrm{-3}\right)$
  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.
  $\mathrm{-5}+3$	$5+\left(\mathrm{-3}\right)$
  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

  $5–3$	$\mathrm{–5}–\left(\mathrm{–3}\right)$
  $2$	$\mathrm{–2}$
  2 positives	2 negatives
  When there would be enough counters of the color to take away, subtract.
  $\mathrm{–5}–3$	$5–\left(\mathrm{–3}\right)$
  $\mathrm{–8}$	$8$
  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
  • $a-b=a+(\mathit{\text{−b}})$
  • $a-\left(\mathrm{-b}\right)=a+b$
• 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 $\mathrm{-1}$
  • Multiplying a number by $\mathrm{-1}$ gives its opposite: $\mathrm{-1}a=-a$
• Division by $\mathrm{-1}$
  • Dividing a number by $\mathrm{-1}$ gives its opposite: $a÷\left(\mathrm{-1}\right)=\mathit{\text{−a}}$

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
  $\text{For any numbers}a,b,c,$ $\text{if}a=b\text{then}a-c=b-c.$	$\text{For any numbers}a,b,c,$ $\text{if}a=b\text{then}a+c=b+c.$

• Division Property of Equality
  • For any numbers $a,b,c,$ and $c\ne 0$
    If $a=b$, then $\frac{a}{c}=\frac{b}{c}$.