Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the first power. The general form is ax + b = c. Solving these equations involves isolating the variable on one side.

• Combine like terms on each side if necessary.

• Move all variable terms to one side and constants to the other.

• Divide or multiply to solve for the variable.

• Check your solution by substituting back into the original equation.

Example: Solve

• Subtract 7 from both sides:

• Divide by 3:

Solving Quadratic Equations

Quadratic equations have the form ax^2 + bx + c = 0. There are several methods to solve them:

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Quadratic Formula:

• Completing the Square: Rewrite the equation in the form and solve for x.

Example: Solve

• Factor:

• Solutions:

Solving Systems of Equations

Systems of equations involve finding values that satisfy two or more equations simultaneously. Common methods include:

• Substitution: Solve one equation for one variable and substitute into the other.

• Elimination: Add or subtract equations to eliminate one variable.

Example: Solve the system:

• Add equations:

• Substitute:

Functions and Their Graphs

Definition of a Function

A function is a relation in which each input (x-value) has exactly one output (y-value). The set of all possible inputs is the domain, and the set of all possible outputs is the range.

• Notation: denotes a function named f with input x.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph more than once.

Example:

Graphing Linear and Quadratic Functions

Graphing functions helps visualize their behavior. Linear functions produce straight lines, while quadratic functions produce parabolas.

• Linear: (slope-intercept form)

• Quadratic:

• Vertex: The highest or lowest point of a parabola, found at

Finding Intercepts

Intercepts are points where the graph crosses the axes.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

Example: For , the y-intercept is and the x-intercept is $2$.

Polynomial and Rational Functions

Polynomial Functions

A polynomial function is a function of the form , where the exponents are whole numbers and coefficients are real numbers.

• Degree: The highest exponent of x.

• Leading Coefficient: The coefficient of the term with the highest degree.

• End Behavior: Determined by the degree and leading coefficient.

Example:

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. This is useful for solving polynomial equations.

• Common Factoring: Factor out the greatest common factor (GCF).

• Factoring Trinomials: Find two numbers that multiply to ac and add to b in .

• Difference of Squares:

Example:

Rational Functions

A rational function is a function of the form , where and are polynomials and .

• Domain: All real numbers except where .

• Vertical Asymptotes: Values of x where .

• Horizontal Asymptotes: Determined by the degrees of and .

Exponential and Logarithmic Functions

Exponential Functions

An exponential function has the form , where , , and .

• Growth: If , the function increases rapidly.

• Decay: If , the function decreases rapidly.

Example:

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and has the form , where , .

• Domain:

• Range: All real numbers

• Key Property:

Analytic Geometry

Distance and Midpoint Formulas

Analytic geometry involves using algebraic methods to solve geometric problems.

• Distance Formula:

• Midpoint Formula:

Example: Find the distance and midpoint between (1,2) and (5,6).

• Distance:

• Midpoint:

Circles

The equation of a circle with center (h, k) and radius r is .

• Center: (h, k)

• Radius: r

Example: has center (2, -3) and radius 4.

Additional Key Concepts

Interval Notation

Interval notation is used to describe sets of numbers, often solutions to inequalities.

• Open interval: (a, b) means all numbers between a and b, not including a or b.

• Closed interval: [a, b] means all numbers between a and b, including a and b.

• Infinity: Use (a, ∞) or (−∞, b) for unbounded intervals.

Example: is written as (2, ∞).

Properties of Functions

Understanding the properties of functions is essential for analyzing their behavior.

• Even Function: (symmetric about the y-axis)

• Odd Function: (symmetric about the origin)

• Increasing/Decreasing: A function is increasing if for .

Summary Table: Methods for Solving Equations

Additional info: Some steps and explanations have been expanded for clarity and completeness based on standard College Algebra curriculum.