Overview
This lecture provides a comprehensive review of essential algebra concepts, including fractions, exponents, equations, and inequalities, to prepare students for upcoming algebra courses.
Fractions: Operations and Simplification
- To add or subtract fractions, convert them to a common denominator before combining numerators.
- When multiplying fractions, multiply numerators together and denominators together; simplify before multiplying if possible.
- To divide fractions, keep the first fraction, change division to multiplication, and flip the second fraction (reciprocal).
Like Terms and Variable Operations
- Only combine like terms (same variable and exponent) when adding or subtracting; e.g., ( 5x - 7x = -2x ).
- To multiply terms with the same base, add exponents: ( x^m \times x^n = x^{m+n} ).
- To divide terms with the same base, subtract exponents: ( x^m / x^n = x^{m-n} ).
- Raising a power to a power multiplies the exponents: ( (x^m)^n = x^{m \times n} ).
- Negative exponents indicate reciprocals: ( x^{-n} = 1/x^n ).
- Any nonzero number raised to the zero power equals 1.
Distribution and Multiplying Polynomials
- Distribute terms by multiplying each part: ( 3x(5x-4) = 15x^2-12x ).
- FOIL method: multiply First, Outer, Inner, Last terms for binomials.
- For larger polynomials, multiply each term and align like terms before combining.
Solving Linear Equations
- Isolate the variable using inverse operations: add/subtract to move constants, multiply/divide to solve for the variable.
- For equations with variables on both sides, group all variables on one side and constants on the other.
- When equations contain fractions or decimals, clear denominators by multiplying through by the least common multiple.
- For equations with two fractions, cross-multiply to solve.
Inequalities and Interval Notation
- When solving inequalities, use similar steps as equations; flip the inequality sign when multiplying/dividing by a negative number.
- Use open circles for strict inequalities (< or >) and closed circles for inclusive inequalities (≤ or ≥) on number lines.
- Write solution sets in interval notation, using parentheses for open intervals and brackets for closed ones.
Key Terms & Definitions
- Like Terms — Terms with identical variable parts and exponents, e.g., ( 2x^2 ) and ( -5x^2 ).
- Reciprocal — Inverse of a fraction; flip numerator and denominator.
- FOIL — Method for multiplying two binomials: First, Outer, Inner, Last.
- Interval Notation — A way to represent solution sets, e.g., (2, ∞) or [1, ∞).
- Distributive Property — ( a(b + c) = ab + ac ).
Action Items / Next Steps
- Practice operations with fractions, combining like terms, and solving equations/inequalities.
- Review textbook sections on exponents, distribution, and interval notation.
- Complete homework assignments related to these topics to reinforce understanding.