Basic Laws of Natural Numbers and Equality

Basic Laws of Natural Numbers

• Closure Law for Addition: For any numbers a and b, their sum a + b results in another number z (e.g., 2 + 5 = 7).
• Commutative Law for Addition: The order of numbers does not affect the sum (e.g., 5 + 7 = 7 + 5 = 12).
• Associative Law for Addition: Grouping of addends does not affect the sum (e.g., 12 + (13 + 22) = (12 + 13) + 22).
• Closure Law for Multiplication: The product of any two numbers a and b is also a number (e.g., 2 x 4 = 8).
• Commutative Law for Multiplication: The order of factors does not affect the product (e.g., 12 x -3 = -3 x 12).
• Associative Law for Multiplication: Grouping of factors does not affect the product (e.g., 6 (-4 x -3) = (6 x -4) x -3).
• Distributive Law: A number multiplied by the sum or difference of two numbers can be distributed (e.g., 3(5 + 8) = 39, 3(5 - 8) = -9).

Basic Laws of Equality

• Reflexive Property: Any number is equal to itself (a = a).
• Symmetric Property: If one number equals another, the reverse is also true (if a = b, then b = a).
• Transitive Property: If a equals b and b equals c, then a equals c.
• Addition of Equals: If a = b and c = d, then a + c = b + d.
• Multiplication of Equals: If a = b and c = d, then ac = bd.

Inequality

• Definition: An inequality compares two quantities indicating one is greater than or less than the other (e.g., a > b).
• Compound Inequality: Represents two inequalities connected by "or" or "and" (e.g., 1 < x ≤ 5).

Theorems of Inequalities

• Theorem 1: Adding the same number to both sides of an inequality maintains the inequality (e.g., if a > b, then a + c > b + c).
• Theorem 2: Multiplying both sides by a positive number preserves the inequality (e.g., if a > b and c > 0, then ac > bc).
• Theorem 3: Multiplying both sides by a negative number reverses the inequality (e.g., if a > b and c < 0, then ac < bc).
• Theorem 4: Changing the signs on both sides of an inequality reverses the inequality (e.g., if a > b, then -a < -b).
• Theorem 5: Taking reciprocals of both sides of a positive or negative inequality also reverses the sense of inequality (e.g., if a > b, then 1/a < 1/b).

Solving Inequalities

• Standard Form: A linear inequality is expressed as ax + b < c. If a is positive, the method to solve is similar to solving equations (e.g., x < (c - b) / a).
• Example Solutions: For the inequality 4x + 7 < 31, solve for x yields x < 6. For -2x + 7 < 23, the solution leads to x > -8.

Compound Inequalities

• Represent two interconnected inequalities. These require understanding how to solve each segment while considering their relationship.