### Key Concepts

### 3.1 Introduction to Integers

- Opposite Notation
- $\xe2\u02c6\u2019a$ means the opposite of the number $a$
- The notation $\xe2\u02c6\u2019a$ is read
*the opposite of*$a.$

- Absolute Value Notation
- The absolute value of a number $n$ is written as $\left|n\right|$.
- $\left|n\right|\xe2\u2030\yen 0$ for all numbers.

### 3.2 Add Integers

**Addition of Positive and Negative Integers**$5+3$ $\mathrm{\xe2\u02c6\u20195}+\left(\mathrm{\xe2\u02c6\u20193}\right)$ 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. $\mathrm{\xe2\u02c6\u20195}+3$ $5+\left(\mathrm{\xe2\u02c6\u20193}\right)$ 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**$5\xe2\u20ac\u201c3$ $\mathrm{\xe2\u20ac\u201c5}\xe2\u20ac\u201c\left(\mathrm{\xe2\u20ac\u201c3}\right)$ $2$ $\mathrm{\xe2\u20ac\u201c2}$ 2 positives 2 negatives When there would be enough counters of the color to take away, subtract. $\mathrm{\xe2\u20ac\u201c5}\xe2\u20ac\u201c3$ $5\xe2\u20ac\u201c\left(\mathrm{\xe2\u20ac\u201c3}\right)$ $\mathrm{\xe2\u20ac\u201c8}$ $8$ 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. **Subtraction Property**- $a\xe2\u02c6\u2019b=a+(\mathit{\text{\xe2\u02c6\u2019b}})$
- $a\xe2\u02c6\u2019\left(\mathrm{\xe2\u02c6\u2019b}\right)=a+b$

**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 negativesPositive

PositiveDifferent Signs Product Positive â€¢ negative

Negative â€¢ positiveNegative

Negative

- To determine the sign of the product of two signed numbers:
**Division of Signed Numbers**- To determine the sign of the quotient of two signed numbers:
Same Signs Quotient Two positives

Two negativesPositive

PositiveDifferent Signs Quotient Positive â€¢ negative

Negative â€¢ PositiveNegative

Negative

- To determine the sign of the quotient of two signed numbers:
**Multiplication by $\mathrm{\xe2\u02c6\u20191}$**- Multiplying a number by $\mathrm{\xe2\u02c6\u20191}$ gives its opposite: $\mathrm{\xe2\u02c6\u20191}a=\xe2\u02c6\u2019a$

**Division by $\mathrm{\xe2\u02c6\u20191}$**- Dividing a number by $\mathrm{\xe2\u02c6\u20191}$ gives its opposite: $a\xc3\xb7\left(\mathrm{\xe2\u02c6\u20191}\right)=\mathit{\text{\xe2\u02c6\u2019a}}$

### 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 $\text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,$

$\text{if}\phantom{\rule{0.2em}{0ex}}a=b\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}a\xe2\u02c6\u2019c=b\xe2\u02c6\u2019c.$$\text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,$

$\text{if}\phantom{\rule{0.2em}{0ex}}a=b\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}a+c=b+c.$**Division Property of Equality**- For any numbers $a,b,c,$ and $c\xe2\u20300$

If $a=b$, then $\frac{a}{c}=\frac{b}{c}$.

- For any numbers $a,b,c,$ and $c\xe2\u20300$