College Algebra Definitions Chapter 1
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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?

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

What is an identity?

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

What is a conditional equation?

<p>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.</p> Signup and view all the answers

What is a contradiction?

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

What is a solution?

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

What is a solution set?

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

What is the graph of an equation?

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

What is solving an equation?

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

What are equivalent equations?

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

What is a simplest equation?

<p>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.</p> Signup and view all the answers

What is a linear equation?

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

What is an inequality?

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

What is a solution of an inequality?

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

What is a solution set of an inequality?

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

What is the graph of an inequality?

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

What are equivalent inequalities?

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

What is a simplest inequality?

<p>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.</p> Signup and view all the answers

What is the quadratic formula?

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

What is a quadratic equation in one variable?

<p>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.</p> 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.

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