Overview
This lecture reviews essential algebra concepts, including operations with fractions, combining like terms, exponents, polynomials, solving equations and inequalities, and interpreting interval notation.
Fractions: Operations and Simplification
- To add or subtract fractions, find a common denominator before combining.
- Multiply fractions by multiplying numerators and denominators directly; simplify when possible.
- When dividing fractions, use "keep, change, flip": multiply by the reciprocal of the second fraction.
Combining Like Terms and Exponent Rules
- Only combine like terms (same variable and exponent) when adding or subtracting in expressions.
- When multiplying terms with the same base, add the exponents.
- When dividing terms with the same base, subtract the exponents.
- Raising a power to another power multiplies the exponents.
- Negative exponents move the variable to the denominator to become positive.
Simplification & Distribution
- Distribute multiplication over addition/subtraction when multiplying a monomial by polynomials.
- Use FOIL (First, Outside, Inside, Last) to multiply two binomials.
- Multiply all terms when expanding binomials or trinomials, then combine like terms.
Solving Linear Equations
- Isolate the variable by reversing operations (addition, subtraction, multiplication, division).
- For equations with variables on both sides, bring variables to one side, constants to the other.
- With fractions or decimals, clear them by multiplying both sides by the denominator or power of 10.
- For two fractions set equal, cross-multiply to solve.
Inequalities and Interval Notation
- Inequalities use open circles for '<' or '>', closed circles for '≤' or '≥' on number lines.
- In interval notation, use parentheses for open intervals, brackets for closed intervals.
- When multiplying/dividing an inequality by a negative, reverse the inequality sign.
Key Terms & Definitions
- Like Terms — terms with the same variable and same exponent.
- FOIL — a method for multiplying two binomials: First, Outside, Inside, Last.
- Interval Notation — a way to write solution sets using brackets and parentheses.
- Cross Multiplication — multiplying across the equal sign to solve equations of two fractions.
Action Items / Next Steps
- Practice provided example problems for each topic.
- Review textbook sections on fractions, exponents, and linear equations.
- Complete relevant homework to reinforce these algebra skills.