Key Concepts
2.1 Use the Language of Algebra
Operation | Notation | Say: | The result is… |
---|---|---|---|
Addition | the sum of and | ||
Multiplication | The product of and | ||
Subtraction | the difference of and | ||
Division | divided by | The quotient of and |
- Equality Symbol
- is read as is equal to
- The symbol is called the equal sign.
- Inequality
- is read is less than
- is to the left of on the number line
- is read is greater than
- is to the right of on the number line
Algebraic Notation | Say |
---|---|
is equal to | |
is not equal to | |
is less than | |
is greater than | |
is less than or equal to | |
is greater than or equal to |
- Exponential Notation
- For any expression is a factor multiplied by itself times, if is a positive integer.
- means multiply factors of
- The expression of is read to the 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 , , and ,
if then
- For any numbers , , and ,
- 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 , , and ,
if then
- For any numbers , , and ,
- 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 |
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 , then and are factors of , and is the product of and .
- 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.
- 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.