An open sentence is neither true nor false until the variables have been replaced by specific values. The process of finding a value for a variable that results in a true sentence is called . This replacement value is called a of the open sentence

Let us see some examples to understand how to solve open sentences.

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Example 1 :

Find the solution of the equation if the replacement set is {10, 11, 12, 13, 14, 15}.

3x - 7 = 29

Solution :

we may decide that the particular value is the solution if it satisfies the given equation.

x = 10

3(10) - 7 = 29

30 - 7  =  29

23  ≠  29 (False)

So, 10 is not the solution of the given equation.

x = 11

3(11) - 7 = 29

33 - 7  =  29

26  ≠  29 (False)

So, 11 is not the solution of the given equation.

x = 12

3(12) - 7 = 29

36 - 7  =  29

29  ≠  29 (True)

So, 12 is the solution of the given equation.

Example 2 :

Find the solution of the equation if the replacement set is {10, 11, 12, 13, 14, 15}.

12(x - 8) = 84

Solution :

We may decide that the particular value is the solution if it satisfies the given equation.

x = 10

12(10 - 8) = 84

12(2)  =  84

24  ≠  84 (False)

So, 10 is not the solution of the given equation.

x = 11

12(11 - 8) = 84

12(3)  =  84

36  ≠  84 (False)

So, 11 is not the solution of the given equation.

x = 12

12(12 - 8) = 84

12(4)  =  84

48  ≠  84 (False)

So, 11 is not the solution of the given equation.

Example 3 :

Find the solution of the equation if the replacement set is {1/4, 1/2, 3/4, 1, 5/4}.

x + (2/5)  =  23/20

Solution :

We may decide that the particular value is the solution if it satisfies the given equation.

x = 1/4

(1/4) + (2/5)  =  23/20

(5 + 8)/20  =  23/20

13/20  ≠  23/20 (False)

So, 1/4 is not the solution of the given equation.

x = 1/2

(1/2) + (2/5)  =  23/20

(5 + 4)/10  =  23/20

9/10  ≠  23/20 (False)

So, 1/2 is not the solution of the given equation.

x = 3/4

(3/4) + (2/5)  =  23/20

(15 + 8)/20  =  23/20

23/20  ≠  23/20 (True)

So, 3/4 is not the solution of the given equation.

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