Open Sentence

Similar to English Sentences, a Mathematical sentence says something.

English Sentence

• The sun rises in the East.

• Ali is Eating.

Mathematical Sentence

• 4 + 4 = 8

• 5 is an odd number.

A sentence can be either open or closed depending on the information provided. Here, we will discuss the meaning of the open sentence in Mathematics.

Open Sentence in Mathematics

An open sentence in Mathematics is neither true nor false until the variables have been substituted by specific values. The method of finding the values of variables that result in a true sentence is known as solving the open sentence. The replacement value is known as the solution of the open sentence.

For example, 1 - y = 8 is an open sentence because the value of ‘y’ is unknown and as a result, we can state if it is true or false.

The uncertainty that we have about 1- y = 8 is what makes 1- y = 8 an open statement.

Open Sentence in Mathematics Example

Q. What will be the solution of the equation if the given replacement set is {10, 11, 12, 13, 14, 15}.

4y - 7 = 29

Solution: We will find the solution of the equation by finding the appropriate value that satisfies the given equation.

Let y = 8

4 (8) - 7  = 29

32 - 7 = 29

25 ≠ 29

So, 8 is not the solution of the given equation

Let y = 9

4 (9) - 7  = 29

36 - 7 = 29

29 = 29

So, 9 is the solution of the given equation.

What are Variables in Open Sentence?

Till now we have learned an open number system that includes a symbol such as ▭ or a 𐩒. Instead of these symbols, we can use variables to represent missing information. These letters are known as variables because their values can change.

In other words, a variable is a symbol that is used for a number we don't know yet. Variables are usually represented by letters such as x or y.

If x = 2, then x + 4 = 2 + 4 = 6.

If x = 3, then x + 4 = 3 + 4 = 7.

If x = 0, then x + 4 = 0 + 4 = 0.

As you can see, the values of x are changing, and that is why it is known as variables.

Solving an Open Sentence with Variables

Variables and open sentences are a part of Mathematics known as Algebra. Open sentences can use a picture symbol (for example a box or a circle) or a variable (a letter denoting any number). Following are some examples of open sentences using variables.

3 + y = 5

x - 4 = -3

p - 17 = 20

2 × n = 16

a ÷ 4 = 16

In these open sentences, we can determine the correct value for the unknown variables that will make the sentence true. This is known as solving an open sentence.

If the sentence that is required to be solved is straightforward, then you can use mental math or your knowledge to solve it.

Solving Open Sentence Using Inverse Operation

An open sentence can also be solved using inverse operations. The inverse operation does the opposite of another operation. Addition and Subtraction are inverse operations, whereas Multiplication and Division are inverse operations.

For example, 7 -  5 is the same as 7 + (- 5). The numbers 5 and - 5 are known as  additive inverse. When we add 5 + (-5), we get the answer 0.

Similarly,  6 ÷ 2/3 is the same as 6 × 3/2. A fraction and its inverse are known as multiplicative inverse. When we multiply, 2/3 × 3/2, we get the answer 1.

Solving Open Sentence Using Inverse Operation Example

Q. Find the value of yin the open sentence:

y +  4 = 16

Solution:

Step 1: We will use the addition and subtraction inverse operation to solve the given open sentence. Subtracting 4 from both the side, we get:

y + 4 - 4 = 16 - 4        (1)

Step 2: Simplify the equation (1) until you get variable = value.

y + 4 - 4 = 16 - 4

y +  0 = 12

y = 12

Hence, the value of y is 12.

Remember: An open is not considered as solved if the variable is negative. For example, - x = 6. The variable should always be positive in the final answer. For example, x = -6.

Solving Open Sentence Using Two Operations

The open sentence may have one or more than one mathematical operations, To solve such an open sentence, you need to remember the correct order given below:

• Bracket

• Of

• Divide

• Multiplication

• Addition 

• Subtraction

Example:

Find the value of y ÷ 2 + 3 = 8

Solution:

Step 1:  Decide which inverse operation should be done first.

As per the BODMAS rule , you must calculate y ÷ 2  before 2 + 3. This implies y ÷ 2  must “ stay together”. You can insert brackets to simplify the problem.

We should use inverse addition first to simplify the sentence as shown below:

(y ÷ 2) + 3 - 3 = 8 - 3

Step 2: Simplify the sentence until you leave with only one operation.

(y ÷ 2) + 3 - 3 = 8 - 3

y ÷ 2  = 5

Step 3: Simplify using the inverse operation until you get the variable = value.

y ÷ 2  = 5

y ÷ 2 × 2 =  5 × 2

y  = 10

We can use the open sentence to solve problems that are described in words. To solve problems, we need to translate the words into mathematical operation as shown below:

Solving Problems Using Open Sentence

Let us now discuss what are open and closed sentences in English.

Open and Closed Sentence in English

In English, a sentence is defined as the basic unit of language that expresses using the basic grammatical rules or syntax. A sentence can either be open or closed depending on the information that is provided.

In English, an open sentence is a sentence that contains whereas a closed sentence is an objective statement that is always true or false.

Example:

Open Sentence - She completed her work - Here ' she' is a variable here.

Closed Sentence - The sun rises in the east (always true).

Closed Sentence - The sun sets in the east (always false).

Solved Examples:

1. Find the value of y in the following open sentence :

y + 6 = 9

Solution:

Step 1: Here, we need to find the value of x that will make the sentence true. To do this, you can ask yourself “ what number should be added to 6 that gives 9?”

As per your knowledge and mental maths, you can add 3 to 6 to get 9.

Step 2: This implies that the value of x is 3. Hence, you can write your answer as x = 3.

2. Find the value of x in the following open number sentence.

y × 4 = 16

Solution:

Step 1: We will use multiplication and division inverse operation to solve the given open sentence. Dividing 4 from both the side, we get,

y × 4 ÷ 4 = 16 ÷ 4       (1)

Step 2: Simplify the equation (1) until you get the format variable = value.

y × 4 ÷ 4 = 16 ÷ 4

y × 1 =  4

y = 4

3. There are 42 students in Class 1. If 20 students are boys, then how many are girls in the class?

Solution: 

Step 1: Understand the given problem.

Total number of  Students = 40

Number of Boys = 20

Number of Girls = x

Step 2: Devise a Plan.

Total number of students = Total number of girls + Total number of boys

42 = 20 + x

Step 3: Execute a Plan.

42 = x + 20

42 - 20 = x + 20 - 20

22 = x

x = 22

Step 4: Look back at the problem.

22 + 20 = 42

An open sentence in math means that it uses variables, meaning that it is not known whether or not the mathematical sentence is true or false. A closed sentence, on the other hand, is a mathematical sentence that is known to be either true or false.

Let us Understand if an Open Sentence Contains a Variable or Not?

Open sentences in mathematics always contain variables. They can be true or false which depends  upon the numbers that are substituted for the variables. Solving open sentences to make them true is an important part of algebra and everyone has to understand the concepts involved in it.

An open sentence or a formula can be true or false for example if x=1, then x+2=3 is true but if the same value x ,x+2=4 is false.

What are the Variables that an Open Sentence Depends on?

The two expressions either use numbers, variables, or a combination of both. A mathematical sentence can also use symbols or words like equals, greater than, or less than. An open sentence in math means that it uses variables, meaning that it is not known whether or not the mathematical sentence is true or false.

In this beginner lesson, students first explore arithmetic sentences to decide whether they are true or false. The lesson then introduces students to sentences that are neither true nor false but are algebraic equations, also called open sentences, such as x + 3 = 7 or 2 x= 12. 

8 + 4 = 5 + 7

5 = 4 + 1

6 • 0 = 6

An open formula does not have a true value.

What is an Open Formula?

An open formula is a formula that contains one variable. It can also have more than 1 but the minimum requirement is1 variable. An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like true or false

Students will first learn the arithmetic sentences to know if it is true or false, even the algebraic expressions are also known as aspen sentences.

As you have seen , the open sentences have many operations to be performed. I am in almost all mathematical calculations. It is very important to learn thoroughly all the concepts with accurate understanding so it becomes easier.

Maths is a subject where all the topics have to be learnt with proper perception and one more thing to be kept in mind is that it cannot be learnt in a day or two. It is slow and a continuous process.

Students should make proper study strategies and practice every day. The more you practice the better the understanding would be.

For all the clarifications you can visit Vedantu’s website and clear all the doubts in the topics. 

About Variables

In mathematics, we can say a variable is a symbol for a quantity that may change or any mathematical object. In particular, we can define a  variable that may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.

A variable can be any characteristic, number or quantity which we can measure. There are mainly three types of variables that are dependent, controlled and independent.

Let Use the Quantitative Variable

A quantitative variable is a variable that can be measured in numeric squares.