# Equations and Inequalities

In mathematics, an equation is a statement that two expressions are equal to each other. The two expressions can be numbers, letters, variables, other equations, or combinations of any of these. The two expressions are said to be equal to each other if one can be substituted for the other in the equation without changing the meaning of the equation. An equation consists of an equal sign (hence the name), and the two expressions on each side of the equal sign. For example, the equation x + 5 = 2 x + 3 is an equation. The symbols + and = are called the "operators" of the equation. The two expressions on each side of the equal sign are called the "arguments" of the equation. The equality symbol, =, is called the "sign" of the equation. Equations are often used as a way of solving a problem by expressing it in terms of numbers. For example, here is how one can solve the equation 3x + 2 = 2x + 1: Simplifying this, 3x + 2 = 2x - 1 The solution to this equation is x = 1/3. The equation x + 5 = 2 x + 3 is an example of the equation: x + 5 = 2 x + 3 The equation x + 5 = 2 x + 3 is the same as the equation: 5 = 2 x + 3 The equation x + 5 = 2 x + 3 is read "5 equals 2 times x plus 3" or "5 equals 2 plus 3 times x". In order to solve an equation, it is necessary to know at least a few rules of operations. The most basic rule of operations is addition. For example, to add numbers x and y, the number x + y is equal to the sum of x and y. An operation that can be used to add numbers is the addition of expressions. For example, the expression x + y is equal to the sum of x and y. In solving an equation, it is necessary to use the rules of operation in the same order. For example, to solve the equation x + 5 = 2 x + 3, we must use the rules in the same order, beginning with the simplest rule, addition. If we begin by adding 5 and 2, the result will be 7. Then, we must add 3 to this result, which will give us the whole number 5. Then, we must add 5 to this, to give us the square of 3, which is 9. Finally, we must add 9 and 3, to give us the square of 9, which is 27. The answer to the equation is 27. Another basic rule is multiplication. For example, to multiply numbers x and y, multiply the numbers xy. An operation that can be used to multiply numbers is the product of expressions. For example, the expression x * y is equal to the product of x and y. In solving an equation, it is necessary to use the rules of operation in the same order. For example, to solve the equation 5 = 2 x + 3, we must use the rules in the same order, beginning with the simplest rule, multiplication. If we multiply 5 and 2, the result will be 10. Then, we must multiply 2 and 3, to give us the product of 5 and 2, which is 10. Then, we must multiply 10 and 3, to give us the product of 10 and 3, which is 30. Finally, we must multiply 30 and 3, to give us the product of 30 and 3, which is 90. The answer to the equation is 90. Sometimes, equations are expressed in the form of an inequality. In an inequality, the left side of the equals sign always contains a variable, and the right side of the equals sign always contains a number. In solving an inequality, it is necessary to use the rules of operation in the same order. For example, to solve the inequality x + 5 < 2 x + 3, we must begin by subtracting 5 and 2 from both sides. The left side is now 5, and the right side is now 3. We must then compare the two sides to see if the inequality is true. Since 5 is less than 3, the inequality is true. The answer to the equation is 3. In addition to the basic rules of operations, there are several other rules of operations that must be followed when solving an equation or inequality. For example, the distributive law is a rule of operations that states that, for any number a and any number b: a * (b + c) = (a * b) + (a * c) The transitive law is a kind of commutative law that states that, for any number a and any number b: a * (b + c) =