Intervals and Sets

This chapter introduces the fundamental concepts of set theory and interval notation, teaching you how to represent number ranges on the number line and perform basic set operations like union and intersection.

Key Concepts

Union

The set of elements that are in either set A or set B (or both).

Set Theory

Intersection

The set of elements common to both sets A and B.

Set Theory

Interval Notation

A concise way to write subsets of the real number line using parentheses and brackets.

Real Numbers

Absolute Value

The distance of a number from zero on the number line, always non-negative.

Arithmetic

🔍 Notation Comparison

Interval Notation	Inequality	Graph Description
[a, b]	a ≤ x ≤ b	Closed circles at both endpoints
(a, b)	a < x < b	Open circles at both endpoints
[a, b)	a ≤ x < b	Closed at a, open at b
(a, b]	a < x ≤ b	Open at a, closed at b

📚 Vocabulary

Union

The set of elements that are in either set A or set B (or both).

Intersection

The set of elements common to both sets A and B.

Interval Notation

A way to describe subsets of the real number line using parentheses and brackets.

Absolute Value

The distance of a number from zero on the number line, always non-negative.

Exponents and Laws of Indices

Master the rules governing powers, including how to multiply, divide, and raise powers to other powers, and how to handle negative and zero exponents.

Key Concepts

Product of Powers

x^m · xⁿ = x^m+n

Laws of Indices

Quotient of Powers

x^m / xⁿ = x^m-n

Laws of Indices

Power of a Power

(x^m)ⁿ = x^{m · n}

Laws of Indices

Negative Exponent

x^-m = 1 / x^m

Algebra

Zero Exponent

x⁰ = 1

Algebra

🛠 Simplification Process

1

Apply Laws: Identify if you are multiplying, dividing, or raising to a power.

2

Combine Exponents: Add, subtract, or multiply exponents accordingly.

3

Handle Negatives: Move terms with negative exponents to the denominator to make them positive.

4

Simplify: Reduce coefficients and final values to their simplest form.

Algebraic Expressions & Polynomials

This chapter defines the building blocks of algebra: constants, variables, and terms. Learn to classify polynomials and perform operations.

Term

A single number, variable, or product of numbers and variables.

Coefficient

The numerical factor of a term.

Constant Term

A term with no variable.

Degree

The highest exponent of the variable in a polynomial.

Like Terms

Terms that have the same variable raised to the same power.

📊 Polynomial Classification

Monomial1 Term (e.g., 5x)

Binomial2 Terms (e.g., x + 2)

Trinomial3 Terms (e.g., x² + 2x + 1)

PolynomialMany Terms

Algebraic Identities & Factorization

Learn to expand expressions using standard identities and reverse the process to factorize polynomials.

📐 Core Identities

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

Factorization

Breaking down an expression into a product of its factors.

Common Factor

A term that is a factor of every term in the polynomial.

Rational Expressions

Covers operations on fractions containing variables, including simplification and finding common denominators.

⚙️ Addition/Subtraction Workflow

1

Find LCD: Determine the Least Common Denominator.

2

Rewrite: Adjust numerators to match the LCD.

3

Combine: Add or subtract the numerators over the common denominator.

4

Simplify: Factor and cancel any common terms.

Equations (Linear & Quadratic)

Solve for unknown variables using linear and quadratic methods, including the quadratic formula and discriminant analysis.

🧮 Essential Formulas

Quadratic Formula:

x = [-b \pm \sqrt(b² - 4ac)] / 2a

Discriminant (D):

D = b² - 4ac

Modeling with Equations

Applying algebraic skills to solve word problems by translating real-world scenarios into mathematical equations.

📝 Word Problem Strategy

1

Define Variable: Identify the unknown (usually x).

2

Set up Equation: Translate text to math (e.g., Area = L * W).

3

Solve: Use algebraic methods to find x.

4

Check: Verify the solution makes sense in context.

Linear Inequalities

Learn to solve inequalities to find ranges of solutions using number line graphing and sign analysis.

🔍 Quadratic Inequality Solving

1

Move Terms: Set the inequality to zero.

2

Factor: Find the roots of the quadratic expression.

3

Find Roots: Mark the roots on a number line.

4

Test Intervals: Check signs between roots to find the solution range.

Coordinate Geometry & Lines

Covers the Cartesian system, distance, midpoints, and finding equations of straight lines.

📐 Geometry Reference

Distance: d = \sqrt[(x₂-x₁)² + (y₂-y₁)²]

Midpoint: M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

Slope (m): m = Δy / Δx

Trigonometry

Introduction to right triangles, angle conversions, and the six trigonometric ratios.

📐 Trigonometric Ratios (SOH CAH TOA)

Sine (sin)

Opposite / Hypotenuse

Cosine (cos)

Adjacent / Hypotenuse

Tangent (tan)

Opposite / Adjacent

Deg ➔ Rad: θ \cdot (π / 180)

Rad ➔ Deg: θ \cdot (180 / π)

Course Assessment

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