Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols
• These symbols represent quantities without fixed values, known as variables
• Algebra is a broad field; fundamental concepts include variables, expressions, equations, and inequalities
• It serves as a foundation for more advanced mathematical topics

Variables

• Variables are symbols (usually letters) representing unknown or changing quantities
• Common examples include x, y, z, a, b, and c
• Variables allow generalization of mathematical relationships
• In the expression "x + 5," x exemplifies a variable which can hold any value

Expressions

• Expressions combine variables, numbers, and mathematical operations (+, -, ×, ÷)
• They do not contain an equals sign (=)
• 3x + 2y - 7 is an example
• Expressions can be simplified by combining like terms
• Like terms have the same variables raised to the same powers

Equations

• Equations are mathematical statements asserting the equality of two expressions
• They contain an equals sign (=)
• The expressions on either side of the equals sign are called the left-hand side (LHS) and the right-hand side (RHS)
• 2x + 3 = 7 is an example equation

Solving Equations

• Solving an equation means finding the value(s) of the variable(s) that make the equation true
• This involves isolating the variable on one side of the equation
• Use inverse operations to maintain equality (e.g., addition/subtraction, multiplication/division)
• To solve 2x + 3 = 7:
  • Subtract 3 from both sides: 2x = 4
  • Divide both sides by 2: x = 2
• The solution x = 2 makes the original equation true

Inequalities

• Inequalities are mathematical statements comparing two expressions using inequality symbols
• Common inequality symbols:
  • < (less than)

  • (greater than)

  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)
• x + 2 > 5 is an inequality example

Solving Inequalities

• Solving inequalities is similar to solving equations, but with some key differences:
  • When multiplying or dividing both sides by a negative number, the inequality sign must be reversed
• If -2x < 6 - Divide both sides by -2 and reverse the inequality sign, resulting in x > -3
• Solutions to inequalities are often expressed as a range of values

Linear Equations

• Linear equations are equations where the highest power of the variable is 1
• They can be written in the form ax + b = c, where a, b, and c are constants
• The graph of a linear equation is a straight line
• Linear equations are fundamental in many areas of mathematics and its applications

Systems of Linear Equations

• A system of linear equations consists of two or more linear equations involving the same variables
• The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously
• Systems can be solved using various methods, including:
  • Substitution: Solve one equation for one variable, then substitute that expression into the other equation
  • Elimination: Add or subtract multiples of the equations to eliminate one variable

Quadratic Equations

• Quadratic equations are equations where the highest power of the variable is 2
• The general form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0
• Can be solved by:
  • Factoring: Express the quadratic expression as a product of two linear factors
  • Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
  • Completing the Square: Manipulating the equation to form a perfect square trinomial
• Quadratic equations can have two, one, or no real solutions

Polynomials

• Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents
• 4x³ - 2x² + x - 5 is an example
• Polynomials can be classified by their degree (the highest exponent of the variable):
  • Linear (degree 1)
  • Quadratic (degree 2)
  • Cubic (degree 3)
  • Quartic (degree 4)
• Polynomials can be added, subtracted, multiplied, and divided

Factoring Polynomials

• Factoring a polynomial involves expressing it as a product of simpler polynomials
• Common factoring techniques include:
  • Factoring out the greatest common factor (GCF)
  • Difference of squares: a² - b² = (a + b)(a - b)
  • Perfect square trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)²
  • Grouping
• Factoring is useful for solving polynomial equations and simplifying expressions

Exponents and Radicals

• Exponents indicate the number of times a base is multiplied by itself
  • Example: x³ = x * x * x
• Rules of exponents:
  • xᵃ * xᵇ = xᵃ⁺ᵇ
  • xᵃ / xᵇ = xᵃ⁻ᵇ
  • (xᵃ)ᵇ = xᵃᵇ
  • (xy)ᵃ = xᵃyᵃ
  • (x/y)ᵃ = xᵃ/yᵃ
  • x⁰ = 1 (if x ≠ 0)
  • x⁻ᵃ = 1/xᵃ
• Radicals (roots) are the inverse operation of exponents
  • The nth root of x is written as ⁿ√x
  • ⁿ√x = y if yⁿ = x
• Rules for radicals:
  • ⁿ√(xy) = ⁿ√x * ⁿ√y
  • ⁿ√(x/y) = ⁿ√x / ⁿ√y

Rational Expressions

• Rational expressions are fractions where the numerator and denominator are polynomials
• (x + 1) / (x² - 4) is an example
• Operations with rational expressions follow the same rules as fractions:
  • Addition/Subtraction: Find a common denominator
  • Multiplication: Multiply numerators and denominators
  • Division: Multiply by the reciprocal of the divisor
• Simplifying rational expressions involves factoring and canceling common factors
• Restrictions on variables: the denominator cannot be zero

Functions

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
• The input is called the argument and the output is called the value of the function
• Functions are often denoted by f(x), where x is the input and f(x) is the output
• The set of all possible inputs is called the domain of the function
• The set of all possible outputs is called the range of the function
• Functions can be represented by:
  • Equations
  • Graphs
  • Tables of values
• Common types of functions:
  • Linear functions: f(x) = mx + b
  • Quadratic functions: f(x) = ax² + bx + c
  • Polynomial functions
  • Rational functions
  • Exponential functions
  • Logarithmic functions

Graphing

• The graph of an equation or function is a visual representation of the relationship between the variables
• In the Cartesian plane (x-y plane):
  • The horizontal axis is the x-axis
  • The vertical axis is the y-axis
  • Points are represented as ordered pairs (x, y)
• Graphing techniques:
  • Plotting points
  • Using intercepts (x-intercept, y-intercept)
  • Using slope and y-intercept (for linear equations)
  • Identifying key features of the function (e.g., vertex of a parabola)