Algebraic Expressions & Inequalities Cheat Sheet

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

Core Principles

• Combine like terms: variables with variables, constants with constants.
• Distributive property: Multiply the term outside the parenthesis by EACH term inside.
• 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.

Action Steps

• Identify like terms (variables and constants).
• Apply the distributive property by multiplying.
• Combine like terms by adding/subtracting coefficients.
• For inequalities, note the comparison symbol (<, >, ≤, ≥).
• When solving inequalities, if multiplying or dividing by a negative, reverse the inequality symbol.
• Check your final answer by substituting values.

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 gives the same result as multiplying each addend by the number and then 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

• Budgeting: Spending 'at most $25' translates to x ≤ 25.
• Minimum Requirements: Needing 'at least $25' translates to x ≥ 25.

Timeline

• Ancient Greece: Early concepts of algebra and symbolic representation emerge.
• 9th Century: Al-Khwarizmi's work introduces systematic algebraic methods.
• 17th Century: Descartes links algebra and geometry; notation develops.
• 19th Century: Formalization of abstract algebra and inequality theory.
• 20th-21st Century: Widespread use in computing, engineering, and data science.

People

• Al-Khwarizmi: Mathematician, often called the 'father of algebra'.
• René Descartes: Philosopher and mathematician, known for Cartesian coordinates linking algebra and geometry.