Key Concepts

2.1 Use the Language of Algebra

OperationNotationSay:The result is…
Additiona+b a plus bthe sum of a and b
Multiplicationa⋅b,(a)(b),(a)b,a(b)a times b The product of a and b
Subtractiona−b a minus b the difference of a and b
Divisiona÷b,a/b,ab,baa 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 NotationSay
a=ba=baa is equal to bb
a≠ba≠baa is not equal to bb
a<ba<baa is less than bb
a>ba>baa is greater than bb
a≤ba≤baa is less than or equal to bb
a≥ba≥baa is greater than or equal to bb

Table2.14

  • 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.
    1. Step 1. Identify like terms.
    2. Step 2. Rearrange the expression so like terms are together.
    3. 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.
    1. Step 1. Substitute the number for the variable in the equation.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. 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.
  • 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.
    1. Step 1. Use the Subtraction Property of Equality to isolate the variable.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. 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.
    1. Step 1. Use the Addition Property of Equality to isolate the variable.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. Step 3. Check the solution.

2.4 Find Multiples and Factors

Divisibility Tests
A number is divisible by
2if the last digit is 0, 2, 4, 6, or 8
3if the sum of the digits is divisible by 3
4if the last two digits are a number divisible by 3
5if the last digit is 5 or 0
6if divisible by both 2 and 3
10if 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.
    1. Step 1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
      1. If the quotient is a counting number, the divisor and quotient are a pair of factors.
      2. If the quotient is not a counting number, the divisor is not a factor.
    2. Step 2. List all the factor pairs.
    3. Step 3. Write all the factors in order from smallest to largest.
  • Determine if a number is prime.
    1. Step 1. Test each of the primes, in order, to see if it is a factor of the number.
    2. Step 2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
    3. 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.
    1. Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
    2. Step 2. If a factor is prime, that branch is complete. Circle the prime.
    3. Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
    4. 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.
    1. Step 1. Divide the number by the smallest prime.
    2. Step 2. Continue dividing by that prime until it no longer divides evenly.
    3. Step 3. Divide by the next prime until it no longer divides evenly.
    4. Step 4. Continue until the quotient is a prime.
    5. 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.
    1. Step 1. List the first several multiples of each number.
    2. 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.
    3. Step 3. Look for the smallest number that is common to both lists.
    4. Step 4. This number is the LCM.
  • Find the LCM using the prime factors method.
    1. Step 1. Find the prime factorization of each number.
    2. Step 2. Write each number as a product of primes, matching primes vertically when possible.
    3. Step 3. Bring down the primes in each column.
    4. Step 4. Multiply the factors to get the LCM.