Contents

Key Concepts

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

Addition of Positive and Negative Integers

$5 + 3 5 + 3$	$−5 + (−3) −5 + (−3)$
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.

Subtraction of Integers

$5 - 3 5 - 3$	$–5 - (–3) –5 - (–3)$
$22$	$–2 –2$
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
- $a - b = a + (−b) a - b = a + (−b)$
- $a - (−b) = a + b a - (−b) = a + b$
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.

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: $−1 a = - a −1 a = - a$
Division by $−1 −1$
- Dividing a number by $−1 −1$ gives its opposite: $a \div (−1) = −a a \div (−1) = −a$

Same Signs	Product
Two positives Two negatives	Positive Positive

Different Signs	Product
Positive • negative Negative • positive	Negative Negative

Same Signs	Quotient
Two positives Two negatives	Positive Positive

Different Signs	Quotient
Positive • negative Negative • Positive	Negative Negative

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
$For any numbers a, b, c, For any numbers$ $if a = b then a - c = b - c . if$	$For any numbers a, b, c, For any numbers$ $if a = b then a + c = b + c . if$

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