Prealgebra

# Key Concepts

PrealgebraKey Concepts

### 3.1Introduction 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.

• 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.

### 3.3Subtract 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.

### 3.4Multiply 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: $a÷(−1)=−aa÷(−1)=−a$

### 3.5Solve 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 $ac=bcac=bc$.