Introduction to Algebra

• Algebra involves using symbols to represent numbers and quantities
• It is used to solve equations and inequalities to find the values of unknown variables

Basic Operations

• Addition combines numbers or expressions to find their sum, such as ( a + b )
• Subtraction finds the difference between two numbers or expressions, such as ( a - b )
• Multiplication finds the product of two or more numbers/expressions, such as ( a \times b ) or ( ab )
• Division splits a number or expression into equal parts, such as ( a \div b ) or ( \frac{a}{b} )

Variables and Constants

• Variables are symbols representing unknown values, for example, ( x, y, z )
• Constants are fixed values that do not change, for example, ( 3, -5, \pi )

Expressions and Equations

• Expressions combine variables, constants, and operations, such as ( 3x + 5 )
• Equations state that two expressions are equal, such as ( 3x + 5 = 14 )

Solving Equations

• Solving equations means finding the value(s) of variables that make the equation true
• Inverse operations are used to isolate the variable
• To solve ( x + 3 = 7 ), subtract 3 from both sides to get ( x = 4 )
• To solve ( 2x = 10 ), divide both sides by 2 to get ( x = 5 )

Linear Equations

• Linear equations can be written as ( ax + b = c ), where ( a, b, ) and ( c ) are constants
• ( x ) is the variable
• The graph of a linear equation is a straight line

Quadratic Equations

• Quadratic equations can be written as ( ax^2 + bx + c = 0 ), where ( a, b, ) and ( c ) are constants and ( a \neq 0 )
• Factoring, completing the square, or using the quadratic formula can solve them
• The quadratic formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

Factoring

• Factoring simplifies an expression into a product of simpler expressions
• To factor ( x^2 + 5x + 6 ), find two numbers that multiply to 6 and add to 5 (2 and 3), resulting in ( (x + 2)(x + 3) )

Systems of Equations

• Systems of equations consist of two or more equations with the same variables
• Substitution, elimination, or graphing can solve them
• Substitution involves solving for one variable and substituting it into the other equation
• Elimination involves adding/subtracting equations to eliminate a variable

Inequalities

• Inequalities compare two expressions using symbols like <, >, ≤, or ≥
• Techniques similar to solving equations are used
• When multiplying/dividing by a negative number, the inequality sign is reversed
• The solution to ( -2x < 6 ) is found by dividing by -2, giving ( x > -3 )

Polynomials

• Polynomials consist of variables and coefficients and use operations of addition, subtraction, multiplication, and non-negative integer exponents
• ( 3x^2 + 2x - 1 ) and ( x^3 - 5x + 7 ) are examples
• Polynomials can be added, subtracted, multiplied, and sometimes divided

Exponents and Radicals

• Exponents indicate how many times a number is multiplied by itself, for example, ( x^3 = x \times x \times x )
• Radicals are the inverse operation of an exponent, for example, ( \sqrt{x} )
• ( x^0 = 1 ) for any non-zero ( x )
• ( x^{-n} = \frac{1}{x^n} )
• ( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} )

Functions

• Functions relate inputs to outputs, where each input has exactly one output
• Written as ( f(x) ), where ( x ) is the input
• ( f(x) = x^2 ) is an example of a function that squares the input
• The domain of a function is the set of all possible input values
• The range is the set of all possible output values

Graphing

• Graphing plots points on a coordinate plane to visualize equations and functions
• The coordinate plane has the x-axis (horizontal) and the y-axis (vertical)
• Linear equations form straight lines, and quadratic equations form parabolas

Logarithms

• Logarithms are the inverse operation to exponentiation
• If ( b^y = x ), then ( \log_b(x) = y )
• Common logarithms use base 10, denoted as ( \log(x) )
• Natural logarithms use base ( e ) (approximately 2.718), denoted as ( \ln(x) )
• Logarithm properties:
  • ( \log_b(mn) = \log_b(m) + \log_b(n) )
  • ( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) )
  • ( \log_b(m^k) = k \times \log_b(m) )

Absolute Value

• Absolute value is a number's distance from zero on the number line
• Denoted as ( |x| ), it's always non-negative
• For instance, ( |-3| = 3 ) and ( |5| = 5 )
• Solving absolute value equations involves considering both positive and negative cases
• For example, solving ( |x - 2| = 3 ) involves considering ( x - 2 = 3 ) and ( x - 2 = -3 ), leading to ( x = 5 ) and ( x = -1 )

Complex Numbers

• Complex numbers are in the form ( a + bi ), with ( a ) and ( b ) as real numbers and ( i ) as the imaginary unit where ( i^2 = -1 )
• ( a ) represents the real part, and ( b ) represents the imaginary part
• Complex numbers can undergo addition, subtraction, multiplication, and division, similar to real numbers, but ( i^2 = -1 )
• Example: ( (3 + 2i) + (1 - i) = 4 + i )