Equations that are TRUE under certain conditions: 
These are the majority of algebraic equations.
Consider x  1 = 9.
This equation has one solution (that makes the equation TRUE) when x = 10, because 10  1 = 9 is true. For all other values of x, the equation is FALSE. Such equations can be referred to as conditional equations because they are TRUE only under certain conditions. For all other values, these equations will be FALSE.

Remember:
When an equation is a linear equation (of degree one), it has only one value as its solution.
The only exceptions are the "strange" situations mentioned in the next two bullets.


Equations that are ALWAYS TRUE: 
Consider x + 7 = 7 + x.
This equation has an infinite number of solutions.
Any value you choose for x will make the equation a TRUE statement. This type of equation is called an identity, and the solution set is all real numbers.
When you solve this type of equation,
the variable will disappear, and you will be left with a TRUE statement about some numeric value. In this example, the "
x" disappears and you are left with the statement, 7 = 7.
A few other examples:
4(x  1) = 4x  4
x + x =
2x
(x + 3)(x  3) = x^{2}  9

Equations that are identities tend to be statements involving "properties" or "rules", such as a property of the real numbers (commutative property, distributive property, etc.), an arithmetic operation on the variable (addition, subtraction, etc), a rule for factoring, and so on. Both sides of the equation represent the same algebraic expression, just written in a different manner. 

If you solve an equation and you end up with an obvious number identity (some number equal to itself),
such as 5 = 5, you'll know that the original equation is also an identity,
with an infinite number of solutions.


Equations that are ALWAYS FALSE: 
Consider x + 7 = x.
This equation has no solutions. No matter what value you choose for
x, the equation will be a FALSE statement. Such statements can be referred to as
contradictions.
The term "
contradiction" refers to statements that are opposed to one another. In relation to equations, it implies that the two sides of the equation are not in agreement, thus no solution will be found.
When you solve this type of equation,
the variable will disappear, and you will be left with a FALSE numerical statement. In this example, the "
x" disappears and you are left with the statement 7 = 0.