1. Introduction to Algebra
- Definition: Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. The symbols (often letters) represent numbers and quantities in formulas and equations.
- Expressions vs. Equations:
- Expression: A combination of numbers, variables, and operations (e.g.,
- Equation: A statement that two expressions are equal (e.g.,
2. Basic Concepts
- Variables: Symbols (typically letters) that represent unknown values. Common examples include , , and .
- Constants: Fixed values, like numbers (e.g., 2, -3, 4.5).
- Coefficients: Numbers that multiply a variable (e.g., in , 3 is the coefficient).
- Terms: Individual parts of an expression, separated by addition or subtraction (e.g., in
and 5 are terms). - Polynomials: Expressions that consist of multiple terms (e.g.,
3. Operations on Algebraic Expressions
- Addition/Subtraction: Combine like terms (e.g., 3x+2x=5x).
- Multiplication: Use distributive property (e.g., ).
- Division: Divide coefficients and subtract exponents for similar bases (e.g., ).
4. Solving Linear Equations
- Linear Equation: An equation of the first degree (e.g.,
- Solution Steps:
- Simplify both sides (if needed).
- Isolate the variable (e.g., move terms involving to one side and constants to the other).
- Solve for the variable (e.g., ).
- Examples:
- → →
- →
5. Inequalities
- Definition: Similar to equations but involve inequalities (, , , ).
- Solving Steps:
- Simplify both sides.
- Isolate the variable.
- Solve, remembering to reverse the inequality sign when multiplying or dividing by a negative number.
- Graphical Representation: Inequalities can be represented on a number line.
6. Factoring
- Definition: Breaking down a complex expression into a product of simpler expressions.
- Common Methods:
- Common Factor: Factor out the greatest common factor (GCF) (e.g.,
- Difference of Squares:
- Quadratic Trinomials:
- Common Factor: Factor out the greatest common factor (GCF) (e.g.,
- Examples:
7. Quadratic Equations
- Definition: Equations of the form .
- Solution Methods:
- Factoring: (if factorable)
- or x = −3
- Quadratic Formula: .
- Completing the Square: Transform the equation into a perfect square trinomial and solve.
8. Exponents and Powers
- Rules:
- (where
- Examples:
- .
9. Radicals and Rational Exponents
- Radicals: Expressions involving roots, such as (square root).
- Properties:
- .
- Rational Exponents:
- Example:
- Example:
10. Functions
- Definition: A relationship between two sets where each input (x) has exactly one output (y).
- Notation: is the function, and is the input.
- Linear Function: (straight-line graph).
- Quadratic Function: (parabola-shaped graph).
11. Systems of Equations
- Definition: A set of two or more equations with the same variables.
- Solution Methods:
- Substitution: Solve one equation for one variable and substitute into the other.
- Elimination: Add or subtract equations to eliminate one variable.
- Graphical Method: Graph both equations and find the point(s) of the intersection.
- Example:
- Steps to Solve:
- Define the variables.
- Write an equation based on the problem.
- Solve the equation.
- Interpret the solution in the context of the problem.
- Examples:
- If a number is doubled and then increased by 5, the result is 15. What is the number? (Equation: ).
13. Rational Expressions and Equations
- Rational Expressions: Fractions where the numerator and/or denominator is a polynomial (e.g., ).
- Operations:
- Addition/Subtraction: Find a common denominator.
- Multiplication: Multiply numerators together and denominators together.
- Division: Multiply by the reciprocal of the divisor.
- Solving Rational Equations: Clear fractions by multiplying by the least common denominator (LCD), then solve the resulting equation.
14. Absolute Value
- Definition: The distance of a number from zero on a number line, is always non-negative.
- if .
- if .
- Absolute Value Equations: Solve by considering both the positive and negative cases.
- Example: leads to
- or ,
- so or
15. Conclusion
Elementary Algebra forms the foundation for more advanced mathematical concepts. These basic principles are crucial for success in higher-level mathematics, including geometry, calculus, and beyond.