Algebra Chapter 1: Tools of Algebra Flashcards

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.