Algebra 2: Chapter 1 Concepts

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Questions and Answers

What are real numbers?

the set of all numbers on the number line

What does 'opposite' refer to on a number line?

numbers that are the same distance from the origin but on opposite sides of it

What is absolute value?

the distance from zero

What is an expression in algebra?

one or more algebraic terms in a phrase Signup and view all the answers

Define an equation.

a sentence formed by placing an equals sign between two expressions Signup and view all the answers

What is an inequality?

a sentence formed by placing an inequality symbol between two expressions Signup and view all the answers

What does the term 'sum' refer to?

the result of adding two or more numbers Signup and view all the answers

What is a difference in mathematics?

the result of subtracting one number from another Signup and view all the answers

Define 'product'.

the product of two or more numbers being multiplied together Signup and view all the answers

What is meant by 'quotient'?

the result of dividing one number by another Signup and view all the answers

What is a power in mathematics?

tell how many times to multiply the base by itself Signup and view all the answers

What is the base in an expression involving powers?

the repeated factor of the power Signup and view all the answers

What does an exponent indicate?

how many times the base occurs as a factor of a power Signup and view all the answers

What are grouping symbols?

symbols used in pairs to group terms together Signup and view all the answers

What does substitution mean in algebra?

to put a value in the place of a variable Signup and view all the answers

What is the order of operations?

the rules that tell you which calculation comes first in an expression Signup and view all the answers

What is a variable?

a symbol used to represent a member of a given set Signup and view all the answers

You _________ an equation.

solve Signup and view all the answers

You ___________ an expression.

simplify Signup and view all the answers

You ___________ an expression by replacing variable with given values and simplifying the result.

evaluate Signup and view all the answers

What is the reflexive property?

a=a Signup and view all the answers

What does the symmetric property state?

if a=b then b=a Signup and view all the answers

What is transitive property in algebra?

if a=b and b=c then a=c Signup and view all the answers

What does the addition property state?

if a=b then a+c=b+c Signup and view all the answers

What does the multiplication property state?

if a=b then ac=bc if c≠0 Signup and view all the answers

What is the distributive property?

a(b+c) = ab + ac Signup and view all the answers

What does the commutative property of addition state?

a+b=b+a Signup and view all the answers

What does the commutative property of multiplication state?

ab=ba Signup and view all the answers

What is the associative property of addition?

(a+b) + c = a + (b+c) Signup and view all the answers

What is the associative property of multiplication?

(ab)c = a(bc) Signup and view all the answers

What does the additive identity property state?

a + 0 = a Signup and view all the answers

What does the multiplicative identity property state?

a ⊗ 1 = a Signup and view all the answers

What is the additive inverse property?

a + (-a) = 0 Signup and view all the answers

What is the multiplicative inverse property?

a (¹/a) = 1 if a≠0 Signup and view all the answers

What does the multiplicative property of zero state?

a ⊗ 0 = 0 Signup and view all the answers

What is an open sentence?

equation or inequality containing a variable Signup and view all the answers

What is a solution (or root) in algebra?

value of a variable that makes a sentence true Signup and view all the answers

What is a solution set?

set of all solutions Signup and view all the answers

What is a null set (empty set)?

set with no members Signup and view all the answers

What does identity refer to in algebra?

solution set is all real numbers Signup and view all the answers

What is a formula?

an equation that states a relationship between two or more variables Signup and view all the answers

Study Notes

Key Concepts in Algebra

Real Numbers: Include all numbers on the number line, encompassing rational and irrational numbers.
Opposite: Refers to numbers equidistant from the origin on a number line but on opposite sides.
Absolute Value: Represents the distance of a number from zero, regardless of direction.
Expression: Comprises one or more algebraic terms grouped in a phrase without an equality statement.
Equation: A mathematical statement that asserts two expressions are equal by using an equals sign.
Inequality: A mathematical sentence that compares two expressions using inequality symbols (e.g., >, <).

Basic Operations

Sum: The result acquired by adding two or more numbers together.
Difference: The result obtained by subtracting one number from another.
Product: The outcome of multiplying two or more numbers.
Quotient: The resulting value when one number is divided by another.

Exponents and Powers

Power: Indicates how many times the base is multiplied by itself.
Base: The number that is repeatedly multiplied when calculating a power.
Exponent: The number that specifies the frequency of the base in multiplication.

Algebraic Structures

Grouping Symbols: Pairs of symbols (e.g., parentheses) used to group terms and prioritize operations in expressions.
Substitution: The act of replacing a variable with a specific value.

Rules of Operations

Order of Operations: A set sequence for performing calculations in expressions to achieve consistent results.
Variable: A symbol that represents an unknown value, often seen in mathematical expressions and equations.

Solving and Simplifying

Solve: The process of finding the value of a variable that satisfies an equation.
Simplify: The act of reducing an expression to its most basic form.
Evaluate: To substitute specific values for variables in an expression and simplify the outcome.

Properties of Equality and Operations

Reflexive Property: Any quantity is equal to itself (e.g., a = a).
Symmetric Property: If one quantity equals another, then the reverse holds true (e.g., if a = b, then b = a).
Transitive Property: Shows the relationship of equality across three quantities (e.g., if a = b and b = c, then a = c).

Additional Mathematical Properties

Addition Property: The equality remains true when the same number is added to both sides of an equation (e.g., if a = b, then a + c = b + c).
Multiplication Property: Similar to the addition property but applies to multiplication (e.g., if a = b, then ac = bc).
Distributive Property: Describes how multiplication distributes over addition (e.g., a(b + c) = ab + ac).
Commutative Properties:
- Addition: The order of addends does not affect the sum (e.g., a + b = b + a).
- Multiplication: The order of factors does not affect the product (e.g., ab = ba).
Associative Properties:
- Addition: Grouping of addends does not affect the sum (e.g., (a + b) + c = a + (b + c)).
- Multiplication: Grouping of factors does not affect the product (e.g., (ab)c = a(bc)).

Identity and Inverses

Additive Identity Property: Adding zero to a number does not change its value (e.g., a + 0 = a).
Multiplicative Identity Property: Any number multiplied by one remains unchanged (e.g., a × 1 = a).
Additive Inverse Property: The sum of a number and its opposite yields zero (e.g., a + (-a) = 0).
Multiplicative Inverse Property: The product of a number and its reciprocal equals one, given the number is not zero (e.g., a × (1/a) = 1).
Multiplicative Property of Zero: Any number multiplied by zero equals zero (e.g., a × 0 = 0).

Sets and Solutions

Open Sentence: An equation or inequality that includes a variable, creating an incomplete statement.
Solution (Root): The specific value of a variable that satisfies an equation or inequality.
Solution Set: The collection of all possible solutions to an equation or inequality.
Null Set (Empty Set): A set that contains no elements.
Identity: In the context of solutions, the solution set includes all real numbers.

Formulas

Formula: A mathematical expression that defines a relationship between two or more variables, often used in various fields of study.

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Test your knowledge of basic algebra concepts with these flashcards from Algebra 2, Chapter 1. Learn definitions for real numbers, opposites, absolute value, and more. Perfect for strengthening your understanding of foundational algebra principles.