College Algebra Definitions Chapter 1

Questions and Answers

What is a unary operation?

A unary operation is a calculation involving one operand.

What is an operand?

An operand is a mathematical object upon which an operator acts.

What is a binary relation?

A binary relation is a comparison of two operands.

What is an equation?

An equation is a mathematical statement which contains an = symbol. Signup and view all the answers

What is an identity?

An identity is an equation which is true for every real number in the domain of the variables. Signup and view all the answers

What is a conditional equation?

A conditional equation is an equation which is true when some real numbers are substituted for the variables and is false when some real numbers are substituted for the variables. Signup and view all the answers

What is a contradiction?

A contradiction is an equation that has no real number solution. Signup and view all the answers

What is a solution?

A solution is a number (or numbers) that makes an equation true when substituted for the variable (or variables). Signup and view all the answers

What is a solution set?

The solution set of an equation is the collection of all solutions of the equation. Signup and view all the answers

What is the graph of an equation?

The graph of an equation consists of all the points, and only those points, whose coordinates are solutions of the equation. Signup and view all the answers

What is solving an equation?

The process of finding all the solutions (the solution set) of an equation is called solving the equation. Signup and view all the answers

What are equivalent equations?

Two equations are equivalent if they have the same solution sets. Signup and view all the answers

What is a simplest equation?

A simplest equation is an equation which has a single variable on one side of the equal sign and a single number on the other side. Signup and view all the answers

What is a linear equation?

A linear equation in one variable is an equation that can be written in the form ax + b = 0. Signup and view all the answers

What is an inequality?

An inequality is a mathematical statement which contains an inequality symbol. Signup and view all the answers

What is a solution of an inequality?

A solution of an inequality is a number (or numbers) that makes an inequality true when substituted for the variable (or variables). Signup and view all the answers

What is a solution set of an inequality?

The solution set of an inequality is the collection of all solutions of the inequality. Signup and view all the answers

What is the graph of an inequality?

The graph of an inequality consists of all the points, and only those points, whose coordinates are solutions of the inequality. Signup and view all the answers

What are equivalent inequalities?

Two inequalities are equivalent inequalities if they have the same solution sets. Signup and view all the answers

What is a simplest inequality?

A simplest inequality is an inequality which has a single variable on one side of the inequality symbol and a single number on the other side. Signup and view all the answers

What is the quadratic formula?

The solutions of a quadratic equation are given by the formula where a, b, and c are the coefficients of the quadratic equation. Signup and view all the answers

What is a quadratic equation in one variable?

A quadratic equation in one variable is an equation which may be written in the form $ax^2 + bx + c = 0$ where a, b, and c are real numbers and a is not zero. Signup and view all the answers

Study Notes

Unary and Binary Operations

A unary operation involves a single operand, essential in functions and calculations.
An operand is the mathematical object that an operator acts upon.
A binary relation compares two operands, illustrating relationships between values.

Equations and Their Types

An equation is a mathematical statement indicated by an '=' symbol, showing equality between two expressions.
An identity is an equation valid for every real number within the variable's domain.
A conditional equation holds true for specific values but may be false for others.
A contradiction has no solution in real numbers; no values satisfy the equation.

Solutions and Sets

A solution is a number that, when substituted into an equation, makes it true.
The solution set encompasses all possible solutions to an equation, forming a complete collection of valid answers.

Graphing Equations

The graph of an equation is defined by all points whose coordinates represent solutions to the equation.
Another phrasing of the graph's definition highlights the importance of satisfying the equation.

Solving Equations

Solving an equation entails finding all solutions, resulting in the comprehensive solution set.

Equations and Inequalities

Equivalent equations have identical solution sets, meaning they yield the same results despite differing forms.
A simplest equation features one variable and one constant, simplifying the solving process.
A linear equation in one variable takes the form ax + b = 0, with 'a' and 'b' being real numbers where not both can be zero.

Understanding Inequalities

An inequality includes an inequality symbol, comparing two expressions similar to equations.
A solution of an inequality makes the inequality true when substituted for its variable(s).
The solution set of an inequality is the full collection of solutions that satisfy the inequality.
The graph of an inequality includes all points satisfying the inequality conditions, indicating valid solutions.

Equivalent Inequalities

Equivalent inequalities possess matching solution sets, allowing for alternative expressions of the same relationship.
A simplest inequality features one variable on one side and one constant on the other, mirroring the format of simplest equations.

Quadratic Concepts

The quadratic formula provides solutions for quadratic equations, defined by coefficients a, b, and c.
A quadratic equation in one variable is structured as ax² + bx + c = 0, with 'a' non-zero, representing a parabolic relation in graphing contexts.

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Explore the fundamental definitions of College Algebra in Chapter 1 through these flashcards. Topics include unary operations, operands, and binary relations. Perfect for reinforcing your understanding of key algebraic concepts.