Open Sentences
An example of an open sentence: x + 3 = 6
First ... what is a "Sentence" ?
Just like an English sentence, in mathematics a sentence says something:
English:
Mathematics:

Now ... what is a "Closed Sentence" or an "Open Sentence" ?
Closed  A closed sentence is always true (or always false). 
Open  A sentence is open when it is not known if it is true or false. 
Examples:
8 is an even number  is closed (it is always true)  
9 is an even number  is closed (it is always false)  
n is an even number  is open (could be true or false, depending on the value of n) 
In that last example:
 if n was 4 the sentence would be true,
 if n was 5 the sentence would be false,
 etc ...
But we didn't say what value n has!
So "n is an even number" may be true or false. So it is open.
Open Sentence
So, we get this definition:
An open sentence can be either true or false depending on what values are used.
Variables
The value we don't know is called a variable (also called an unknown)
In this example of an open sentence, x is a variable:
x + 3 = 8
In this example, w and q are both variables:
w + q = 2
Solving
Solving means finding a value for the variable that makes the sentence true.
Example: Solve x + 3 = 8
Let us subtract 3 from both sides:
x + 3  3 = 8  3
x = 5
Check: 5 + 3 = 8 is true
So we have solved x + 3 = 8 by making x = 5
Some More Examples
Here are some more examples of closed and open sentences for you:
Closed Sentences:
A square has four corners  always true  
6 is less than 5  always false  
3 is a negative number  always true 
Open Sentences:
A triangle has n sides  Can be true or false (depending on the value of n)  
z is a positive number  Can be true or false (depending on the value of z)  
3y = 4x + 2  Can be true or false (depending on the values of x and y)  
a + b = c + d  Can be true or false (depending on the values of a,b,c,d) 