Properties of Equations and Inequalities

Algebraic Equations and Inequalities

Basic Conceptions

Properties of Equations and Inequalities

Graphical Interpretation of Solutions

Linear Equations and Inequalities

Linear Equations

Linear Inequalities

Linear Equations and Inequalities Involving Absolute Values

Linear Equations Involving Absolute Value

Linear Equations Involving a Few Absolute Values

Linear Inequalities Involving Absolute Value

Quadratic Equations and Inequalities

Quadratic Equations
Quadratic Equations and Quadratic Functions

Extreme Value of Quadratic Function

Quadratic Formula

Solving Quadratic Equations by Factoring

Quadratic Inequalities

Properties of Equations and Inequalities

Key Topics Remaining: Linear Equations » Linear Inequalities » Linear Equations Involving Absolute Values » Linear Inequalities Involving Absolute Values » Quadratic Equations » Quadratic Equations and Quadratic Functions » Extreme Value of Quadratic Function » Quadratic Formula » Solving Quadratic Equations by Factoring » Quadratic Inequalities

The term "equation" is used to assert the equality of two quantities.
An algebraic equation states that one algebraic expression is equal to another.
A statement that two quantities, a and b, are equal, indicated by the symbolical expression

a = b,

no matter whether a and b are numbers or algebraic expressions.
However, there are essential distinctions between the equality of numbers and the equality of algebraic expressions.

If a and b are two numbers, then a mathematical statement a = b is either true or false. For example, the statement

is true, while the equality
asserts a false proposition.
If a and b are algebraic expressions involving a variable, then a mathematical statement a = b can be true (for some set of values of the variable) or false (for other values). For instance, the equality 2 x = 8 is true for x = 4, while it is false for all other values of x .

To solve an equation means to find all values of the variable, which reduce the equation to an identity.

The set of all such values is called the solution set of the equation.
A solution of an equation is also called a root of the equation.

Equations are called equivalent, if they have the same solution sets.
Some equations have no solutions, that is, the result of substituting any numbers for the variables is a false statement. Equations of such a kind are called inconsistent.

It is impossible to formulate a well-defined algorithm of solving an arbitrary algebraic equation except for some simplest types of equations such as linear or quadratic equations.
To solve a more complicated problem, it is necessary to simplify the equation as much as possible, having in view to transform a given problem to a standard pattern.

The properties of equations are based on the properties of real numbers.

a = b if and only if a + c = b + c for any c .

a = b if and only if a c = b c for any c .

The first property allows to add a quantity to both sides of an equation, asserting that the obtained equation is equivalent to the given one.

The second property states that the multiplication of both sides of an equation by any non-zero quantity results in an equation being equivalent to the given one.

To check up whether a number is a solution of an equation, substitute the number for the variable in the equation and see if the result is an identity.