Algebraic Properties & Inequalities Cheat Sheet

Master the distributive property for combining terms and understand the nuances of inequality symbols and their manipulation, especially when dealing with negative numbers.

Core Principles

• Distributive Property: Multiply each term inside parentheses by the factor outside.
• Combine Like Terms: Add or subtract coefficients of variables, and combine constant terms separately.
• Cannot Combine Unlike Terms: Variables with different powers or constants cannot be combined with each other.
• Order of Operations: Respect the order of operations; do not simplify outside parentheses before distributing.
• Inequality vs. Equation: Equations use '=', inequalities use '>', '<', '≥', '≤'.
• Inequality Symbols: '≥' means greater than or equal to; '≤' means less than or equal to.
• Interpreting Phrases: 'Most' implies '≤'; 'At least' implies '≥'.
• Dividing by Negative: Dividing (or multiplying) an inequality by a negative number reverses the inequality sign.

Action Steps

• Identify terms inside and outside parentheses.
• Multiply the outside factor by each inside term (distribute).
• Group terms with the same variable.
• Group constant terms.
• Combine coefficients of like variables.
• Combine constant terms.
• For inequalities, identify the inequality symbol.
• When solving, if multiplying or dividing by a negative number, reverse the inequality symbol.

Formulas

• Distributive Property: a(b + c) = ab + ac
• Combining like terms: ax + bx = (a+b)x
• Combining constants: c + d

Key Terms

• Distributive Property: A rule stating that multiplying a sum by a number is the same as multiplying each addend by the number and adding the products.
• Coefficient: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
• Like Terms: Terms that have the same variable(s) raised to the same power(s).
• Inequality: A mathematical statement that compares two expressions using symbols like >, <, ≥, or ≤.
• Equation: A mathematical statement that asserts the equality of two expressions, using the '=' symbol.

Real World Examples

• Budgeting for a purchase.: If the most you can spend is $25, the cost 'c' must satisfy c ≤ 25.
• Meeting a minimum requirement.: If you need to score at least 70 on a test, your score 's' must satisfy s ≥ 70.

Timeline

• Ancient Mesopotamia: Early forms of algebraic problem-solving emerge.
• Ancient Greece: Euclid's Elements introduces systematic geometric proofs and algebraic concepts.
• 9th Century: Al-Khwarizmi's 'The Compendious Book on Calculation by Completion and Balancing' formalizes algebraic methods.
• 16th Century: Development of symbolic algebra with letters representing unknowns and constants.
• 17th Century: Descartes links algebra and geometry with coordinate systems; Newton develops calculus.
• 19th Century: Abstract algebra develops, exploring structures beyond numbers.