Algebra Flashcards for Chapter 1: Tools of Algebra

Questions and Answers

What is the opposite of a number?

The number itself

The sum of the number and its additive inverse is zero (correct)

The distance of the number from zero

The product of the number and its multiplicative inverse is one

What is the definition of a reciprocal?

The product is 1.

What is the absolute value of a number?

The distance of a number from zero on the number line.

What is a variable?

A symbol, usually a letter, that represents one or more numbers. Signup and view all the answers

What is an algebraic expression?

An expression that contains one or more variables. Signup and view all the answers

What does it mean to evaluate an expression?

Substituting numbers for the variables in an expression and simplifying it. Signup and view all the answers

Define a term in algebra.

A number, variable, or the product of a number and one or more variables. Signup and view all the answers

What is a coefficient?

The number factor in a term. Signup and view all the answers

What is a solution in algebra?

A number that makes the equation true. Signup and view all the answers

What is an extraneous solution?

A solution derived from an original equation that is not valid. Signup and view all the answers

What is an equation?

A mathematical statement that two expressions are equal. Signup and view all the answers

What is the order of operations?

PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Signup and view all the answers

What are natural numbers?

The set of numbers 1, 2, 3, 4,... Signup and view all the answers

What are whole numbers?

Natural numbers including zero. Signup and view all the answers

What are integers?

All whole numbers both positive and negative, including zero. Signup and view all the answers

What are rational numbers?

Any number that can be expressed as a fraction. Signup and view all the answers

What are irrational numbers?

Numbers that cannot be expressed in the form a/b, where a and b are integers. Signup and view all the answers

What are the properties of real numbers?

Commutative, Associative, Identity, Inverse, Distributive. Signup and view all the answers

What is the commutative property?

a + b = b + a and ab = ba. Signup and view all the answers

What is the associative property?

(a + b) + c = a + (b + c) or (ab)c = a(bc). Signup and view all the answers

What is the identity property?

a * 1 = a and a + 0 = a. Signup and view all the answers

What is the inverse property?

a + (-a) = 0 and a × (1/a) = 1. Signup and view all the answers

What is the distributive property?

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

What are compound inequalities?

Two inequalities joined by the word 'and' or 'or'. Signup and view all the answers

What are inequalities?

A mathematical sentence involving <, >, ≤, or ≥. Signup and view all the answers

Study Notes

Algebra Tools Overview

Opposite: The additive inverse of a number; e.g., 3 + (-3) = 0.
Reciprocal: The multiplicative inverse; the product of a number and its reciprocal equals 1.

Fundamental Concepts

Absolute Value: Represents the distance of a number from zero on the number line.
Variable: A symbol, often a letter, that stands for one or more numbers.

Expressions and Evaluations

Algebraic Expression: Contains one or more variables; examples include 2x + 3 and z - y.
Evaluate: To substitute values for variables and simplify using the order of operations; e.g., evaluating 2x + 3y for x = 2 and y = -1 gives 1.

Terms and Equations

Term: A number, variable, or a product of a number and variables; identified in expressions.
Coefficient: The numerical factor in a term, determining the term's value.
Solution: A value that satisfies an equation; e.g., in x = 6, 6 is the solution.
Extraneous Solution: A solution derived from an equation that does not satisfy the original equation.

Mathematical Statements

Equation: A statement asserting two expressions are equal, such as 2x = 4.
Order of Operations: Rules for evaluating expressions in the order: parentheses, exponents, multiplication/division, addition/subtraction (PEMDAS).

Number Sets

Natural Numbers: Counting numbers beginning from 1; does not include zero.
Whole Numbers: Natural numbers plus zero (0, 1, 2, 3,...).
Integers: All whole numbers, including positive, negative, and zero.
Rational Numbers: Numbers expressible as fractions; e.g., ¾.
Irrational Numbers: Numbers that cannot be represented as a fraction of integers; examples include √2 and non-repeating decimals.

Properties of Real Numbers

Commutative Property: States a + b = b + a and ab = ba; the order of addition or multiplication does not affect the result.
Associative Property: Shows how numbers are grouped; (a+b)+c = a+(b+c) or (ab)c = a(bc).
Identity Property: Adds zero or multiplies by one without changing values; e.g., a + 0 = a and a * 1 = a.
Inverse Property: Defines how numbers can cancel each other; a + (-a) = 0 and a × (1/a) = 1 (where a ≠ 0).
Distributive Property: Relates multiplication to addition; e.g., a(b+c) = ab + ac.

Inequalities

Compound Inequalities: Combine two inequalities with "and" or "or".
Inequalities: Mathematical sentences using symbols such as <, >, ≤, and ≥ to compare values.

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Description

This quiz focuses on key terms and definitions from Chapter 1 of Algebra. Through flashcards, you'll learn important concepts such as opposites, reciprocals, absolute values, and variables. Perfect for reinforcing your understanding of basic algebraic tools.

Algebra Chapter 1: Tools of Algebra Flashcards