Overview
This lecture reviews essential algebra concepts such as working with fractions, combining like terms, operations with exponents, equation solving, inequalities, and introduces interval notation.
Operations with Fractions
- To add/subtract fractions, get a common denominator, adjust numerators, then add or subtract.
- To multiply fractions, multiply numerators and denominators directly; reduce if possible.
- To divide fractions, "keep-change-flip": multiply the first fraction by the reciprocal of the second.
- Simplify before multiplying large fractions to avoid big numbers.
Combining Like Terms and Distributive Property
- Only add/subtract terms with exactly the same variables and exponents (like terms).
- To multiply terms with the same base, add exponents; to divide, subtract exponents.
- Distribute a monomial over a polynomial by multiplying it with each term inside.
Exponent Rules
- Multiplying variables: add exponents (e.g. x⁴·x⁷ = x¹¹).
- Dividing variables: subtract exponents (x⁷/x⁴ = x³).
- Raising a power to a power: multiply exponents ((x³)⁴ = x¹²).
- Negative exponents: x⁻³ = 1/x³; always express exponents positively in final answers.
- Anything to the zero power equals one (x⁰ = 1).
Expanding and Multiplying Polynomials
- FOIL is used to multiply two binomials: first, outside, inside, last terms.
- When multiplying binomials or trinomials, distribute each term from one to every term of the other.
- Combine like terms in the result.
Solving Linear Equations
- Isolate the variable by performing inverse operations (addition/subtraction, then multiplication/division).
- Multi-step equations: undo addition/subtraction before multiplication/division.
- With variables on both sides, gather like terms and isolate the variable.
Equations with Fractions or Decimals
- Eliminate fractions by multiplying both sides by the common denominator.
- Eliminate decimals by multiplying by a power of 10.
- Cross-multiply to solve equations where both sides are single fractions.
Inequalities and Interval Notation
- Use open circles for '<' or '>'; closed circles for '≤' or '≥' when graphing on number lines.
- When multiplying/dividing both sides by a negative, reverse the inequality sign.
- Interval notation: use parentheses for open, brackets for closed intervals, and always use parentheses for infinity.
Key Terms & Definitions
- Like Terms — Terms with the same variables and exponents.
- Exponent — The number that indicates how many times a base is multiplied by itself.
- FOIL — First, Outside, Inside, Last; a method for multiplying two binomials.
- Interval Notation — A way to describe solution sets using parentheses and brackets.
- Reciprocal — The flipped version of a fraction (a/b becomes b/a).
Action Items / Next Steps
- Practice example problems for each concept, including fraction operations, combining like terms, and solving equations.
- Review homework from your textbook for additional practice.
- Prepare for quizzes on fractions, equations, and inequalities.