**Solving open sentences :**

A mathematical statement with one or more variables is called an open sentences.

An open sentence is neither true nor false until the variables have been replaced by specific values.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.

**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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