**2.1** Use the Language of Algebra

Operation | Notation | Say: | The result is… |
---|---|---|---|

Addition | a+b | a plus b | the sum of a and b |

Multiplication | a⋅b,(a)(b),(a)b,a(b) | a times b | The product of a and b |

Subtraction | a−b | a minus b | the difference of a and b |

Division | a÷b,a/b,ab,ba | a divided by b | The quotient of a and b |

**Equality Symbol**- a=ba=b is read as aa is equal to bb
- The symbol == is called the equal sign.

**Inequality**- a<ba<b is read aa is less than bb
- aa is to the left of bb on the number line
- a>ba>b is read aa is greater than bb
- aa is to the right of bb on the number line

Algebraic Notation | Say |
---|---|

a=ba=b | aa is equal to bb |

a≠ba≠b | aa is not equal to bb |

a<ba<b | aa is less than bb |

a>ba>b | aa is greater than bb |

a≤ba≤b | aa is less than or equal to bb |

a≥ba≥b | aa is greater than or equal to bb |

**Table****2.14**

**Exponential Notation**- For any expression anan is a factor multiplied by itself nn times, if nn is a positive integer.
- anan means multiply nn factors of aa
- The expression of anan is read aa to the nthnth power.

**Order of Operations** When simplifying mathematical expressions perform the operations in the following order:

- Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
- Exponents: Simplify all expressions with exponents.
- Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
- Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

**2.2** Evaluate, Simplify, and Translate Expressions

**Combine like terms.**- Step 1. Identify like terms.
- Step 2. Rearrange the expression so like terms are together.
- Step 3. Add the coefficients of the like terms

**2.3** Solving Equations Using the Subtraction and Addition Properties of Equality

**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.

**Subtraction Property of Equality**- For any numbers aa, bb, and cc,ifa=ba=bthena−c=b−ca−c=b−c

**Solve an equation using the Subtraction Property of Equality.**- Step 1. Use the Subtraction Property of Equality to isolate the variable.
- Step 2. Simplify the expressions on both sides of the equation.
- Step 3. Check the solution.

**Addition Property of Equality**- For any numbers aa, bb, and cc,ifa=ba=bthena+c=b+ca+c=b+c

**Solve an equation using the Addition Property of Equality.**- Step 1. Use the Addition Property of Equality to isolate the variable.
- Step 2. Simplify the expressions on both sides of the equation.
- Step 3. Check the solution.

**2.4** Find Multiples and Factors

Divisibility Tests | |
---|---|

A number is divisible by | |

2 | if the last digit is 0, 2, 4, 6, or 8 |

3 | if the sum of the digits is divisible by 3 |

4 | if the last two digits are a number divisible by 3 |

5 | if the last digit is 5 or 0 |

6 | if divisible by both 2 and 3 |

10 | if the last digit is 0 |

**Factors**If a⋅b=ma⋅b=m, then aa and bb are factors of mm, and mm is the product of aa and bb.**Find all the factors of a counting number.**- Step 1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
- If the quotient is a counting number, the divisor and quotient are a pair of factors.
- If the quotient is not a counting number, the divisor is not a factor.

- Step 2. List all the factor pairs.
- Step 3. Write all the factors in order from smallest to largest.

- Step 1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
**Determine if a number is prime.**- Step 1. Test each of the primes, in order, to see if it is a factor of the number.
- Step 2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
- Step 3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

**2.5** Prime Factorization and the Least Common Multiple

**Find the prime factorization of a composite number using the tree method.**- Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
- Step 2. If a factor is prime, that branch is complete. Circle the prime.
- Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
- Step 4. Write the composite number as the product of all the circled primes.

**Find the prime factorization of a composite number using the ladder method.**- Step 1. Divide the number by the smallest prime.
- Step 2. Continue dividing by that prime until it no longer divides evenly.
- Step 3. Divide by the next prime until it no longer divides evenly.
- Step 4. Continue until the quotient is a prime.
- Step 5. Write the composite number as the product of all the primes on the sides and top of the ladder.

**Find the LCM by listing multiples.**- Step 1. List the first several multiples of each number.
- Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
- Step 3. Look for the smallest number that is common to both lists.
- Step 4. This number is the LCM.

**Find the LCM using the prime factors method.**- Step 1. Find the prime factorization of each number.
- Step 2. Write each number as a product of primes, matching primes vertically when possible.
- Step 3. Bring down the primes in each column.
- Step 4. Multiply the factors to get the LCM.

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