Properties of equations and inequalities are fundamental concepts in mathematics. When comparing two numbers, we have several basic methods. The subtraction method compares the difference a minus b with zero. If a minus b is positive, then a is greater than b. The division method compares the ratio a divided by b with one, which is useful when both numbers are positive.
The subtraction method is the most fundamental approach for comparing numbers. The process involves four clear steps: first, calculate the difference a minus b; second, simplify the expression; third, compare the result with zero; and fourth, draw your conclusion. For example, to compare 3 and 1, we calculate 3 minus 1 equals 2. Since 2 is greater than 0, we conclude that 3 is greater than 1.
The division method is particularly useful when comparing two positive numbers. The process follows four steps: first, calculate the ratio a divided by b; second, simplify the expression; third, compare the result with 1; and fourth, draw your conclusion. For example, to compare 6 and 2, we calculate 6 divided by 2 equals 3. Since 3 is greater than 1, we conclude that 6 is greater than 2.
Inequalities have several fundamental properties that are essential for mathematical reasoning. The symmetry property states that if a is greater than b, then b is less than a. The transitivity property shows that if a is greater than b and b is greater than c, then a must be greater than c. We can add the same number to both sides of an inequality without changing the relationship. For multiplication, we must be careful: multiplying by a positive number preserves the inequality direction, but multiplying by a negative number reverses it.