Algebraic Expressions & Inequalities Cheat Sheet

Master the distributive property for combining terms and understand the nuances of inequality signs, including when and how to flip them.

Core Principles

• Combine like terms: variables with variables, constants with constants.
• Distributive property: Multiply the outside term by each term inside the parentheses.
• Treat subtraction as adding a negative.
• Inequalities use <, >, ≤, ≥ symbols; equations use =.
• Flipping the inequality sign is crucial when multiplying or dividing by a negative number.
• Understand 'at most' (≤) and 'at least' (≥) for real-world inequality setup.

Action Steps

• Identify like terms (variables and constants).
• Apply the distributive property by multiplying.
• Combine like terms after distribution.
• For inequalities, check if multiplying or dividing by a negative.
• If dividing/multiplying by a negative in an inequality, flip the sign.
• Translate word problems into inequalities using correct symbols (≤, ≥).

Formulas

• Distributive Property: $a(b + c) = ab + ac$
• Distributive Property with Negative: $-a(b + c) = -ab - ac$

Key Terms

• Coefficient: A numerical factor multiplying a variable.
• Constant: A number without a variable.
• Distributive Property: Multiplying a sum by a number is the same as multiplying each addend by the number and adding the products.
• Inequality: A mathematical statement comparing two expressions using <, >, ≤, or ≥.
• Equation: A mathematical statement that two expressions are equal, using the = sign.

Real World Examples

• Spending limit: The most I can spend is $25 means: Amount Spent ≤ $25.
• Minimum requirement: I need at least $25 means: Amount Needed ≥ $25.

Timeline

• Ancient Greece: Early concepts of algebra and equations explored.
• 9th Century: Al-Khwarizmi's work formalizes algebraic methods.
• 17th Century: Descartes introduces notation for inequalities.
• 19th Century: Development of abstract algebra and formal inequality theory.
• 20th Century: Widespread use in calculus, optimization, and computer science.

People

• Al-Khwarizmi: Often called the 'father of algebra' for his systematic approach.
• René Descartes: Introduced modern notation for inequalities (<, >).

Quiz

• What happens to the inequality sign when dividing by -2?: It flips
• Which symbol represents 'at most 50'?: ≤ 50