Understanding the Domination Laws | Manipulating Inequalities in Mathematics

Domination laws

In mathematics, the domination laws refer to a set of properties that apply to inequalities involving real numbers

In mathematics, the domination laws refer to a set of properties that apply to inequalities involving real numbers. They provide rules for manipulating and simplifying inequalities in various ways. These laws are important for solving equations and inequalities, as well as for proving mathematical statements.

The domination laws are as follows:

1. Reflexive Law: For any real number a, a ≤ a (a is less than or equal to itself). This law simply states that any number is always less than or equal to itself.

2. Transitive Law: For any real numbers a, b, and c, if a ≤ b and b ≤ c, then a ≤ c. This law means that if one number is less than or equal to another, and the second number is less than or equal to a third number, then the first number is also less than or equal to the third number.

3. Addition Law: For any real numbers a, b, and c, if a ≤ b, then a + c ≤ b + c. This law states that if one number is less than or equal to another, then adding the same value to both numbers does not change the inequality.

4. Subtraction Law: For any real numbers a, b, and c, if a ≤ b, then a – c ≤ b – c. This law is similar to the addition law, but instead of adding a number, we subtract the same value from both sides of the inequality.

5. Multiplication Law: For any real numbers a, b, and c, if a ≤ b and c > 0, then ac ≤ bc. This law states that if one number is less than or equal to another, and we multiply both numbers by a positive number, the inequality remains true.

6. Division Law: For any real numbers a, b, and c, if a ≤ b and c > 0, then a/c ≤ b/c. This law is similar to the multiplication law, but instead of multiplying, we divide both sides of the inequality by a positive number.

These domination laws are powerful tools for manipulating and simplifying inequalities, allowing us to compare and solve mathematical expressions involving real numbers. By applying these laws correctly, we can establish relationships between numbers and prove various mathematical statements.

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