Introduction to Basic Algebra

• Basic algebra deals with symbols and rules for manipulating them.
• It generalizes arithmetic by extending numbers and operations to variables and algebraic expressions.

Variables

• Variables are symbols, usually letters, representing unknown or changeable values.
• They express relationships and facilitate general problem-solving.
• In "3x + 5," x is a variable.

Constants

• Constants are fixed values within an expression.
• In "3x + 5," 3 and 5 are constants.

Algebraic Expressions

• Algebraic expressions combine variables, constants, and math operations (+, -, ×, ÷).
• They do not contain an equals sign (=).
• Examples include 3x + 5, 2y - 7, and a^2 + b^2.

Equations

• Equations show the equality between two algebraic expressions.
• They include an equals sign (=).
• Examples include 3x + 5 = 14, 2y - 7 = 9, and a^2 + b^2 = c^2.

Terms

• Terms are components of an algebraic expression, separated by + or - signs.
• In "3x + 5," 3x and 5 are considered terms.

Coefficients

• Coefficients are numerical parts of a term that includes a variable.
• In "3x," 3 is the coefficient.

Basic Operations

• The basic operations in algebra include addition, subtraction, multiplication, and division.
• These operations are used to manipulate algebraic expressions and equations.

Addition

• Combining like terms involves adding their coefficients.
• For example: 3x + 2x = 5x

Subtraction

• Similar to addition, subtraction combines like terms by subtracting coefficients.
• For example: 5y - 2y = 3y

Multiplication

• Multiplying algebraic expressions means distributing each term of one expression to each term of the other.
• For example: 2(x + 3) = 2x + 6

Division

• Dividing algebraic expressions may involve simplifying fractions or using long division.
• For example: (6x + 9) / 3 = 2x + 3

Order of Operations

• PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) dictates the sequence of operations.
• The order ensures consistency when evaluating expressions: Parentheses/brackets first, then exponents/roots, multiplication/division (left to right), and addition/subtraction (left to right).
• For example: 2 + 3 × 4 = 2 + 12 = 14 (multiplication occurs before addition).

Simplifying Expressions

• Simplifying includes combining like terms, applying the distributive property, and factoring.

Combining Like Terms

• Like terms have the same variable raised to the same power.
• Combine like terms by adding or subtracting their coefficients.
• For example: 3x^2 + 2x^2 - x + 4x = 5x^2 + 3x

Distributive Property

• The distributive property is a(b + c) = ab + ac.
• It removes parentheses in algebraic expressions.
• For example: 4(2x - 5) = 8x - 20

Factoring

• Factoring reverses the distributive property.
• Factoring expresses an algebraic expression as a product of its factors.
• For example: 6x + 9 = 3(2x + 3)

Solving Equations

• Solving equations means finding variable values that make the equation true.
• Use inverse operations to isolate the variable.

Linear Equations

• Linear equations have a highest variable power of 1.
• They are written as ax + b = c, where a, b, and c are constants.
• Solve by isolating the variable using inverse operations, adding/subtracting constants, and dividing/multiplying by the variable's coefficient.
• Example: Solve 2x + 3 = 7
• 2x = 7 - 3
• 2x = 4
• x = 4 / 2
• x = 2

Quadratic Equations

• Quadratic equations have a highest variable power of 2.
• They are written as ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Solve by factoring, completing the square, or using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
• Factoring expresses the quadratic expression as a product of two binomials.
• Completing the square manipulates the equation to form a perfect square trinomial.
• Example: Solve x^2 - 5x + 6 = 0
• (x - 2)(x - 3) = 0
• x = 2 or x = 3

Systems of Equations

• These are sets of two or more equations with the same variables.
• Solving involves finding variable values that satisfy all equations simultaneously via substitution, elimination, or graphing.
• Substitution solves one equation for a variable and substitutes that expression into another equation.
• Elimination adds or subtracts equations to eliminate a variable.
• Graphing plots equations to find intersection points.
• Example: Solve the system of equations:
• x + y = 5
• x - y = 1
• Adding the two equations gives:
• 2x = 6
• x = 3
• Substituting x = 3 into the first equation gives:
• 3 + y = 5
• y = 2
• Therefore, the solution is x = 3 and y = 2.

Inequalities

• Inequalities compare expressions using symbols like <, >, ≤, and ≥.
• Solving finds the range of variable values that make the inequality true.

Linear Inequalities

• These are inequalities with a highest variable power of 1.
• They are written as ax + b < c, ax + b > c, ax + b ≤ c, or ax + b ≥ c, where a, b, and c are constants.
• Solving is like solving linear equations, except multiplying/dividing by a negative number reverses the inequality sign.
• Example: Solve 3x - 2 > 7
• 3x > 9
• x > 3

Absolute Value

• A number's absolute value is its distance from zero on the number line, denoted |x|.
• Absolute value is always non-negative.
• Example: |-5| = 5 and |5| = 5