# Open Sentences and Identities

The term open sentences refer to a type of sentence that is used in maths. A sentence in maths is just the same as a sentence in English;

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$2+2=4$

$5×5=25$

## Open sentences definition in maths

An open sentence is a sentence where it is not known whether it is true or false until the missing variables have been solved, whereas a closed sentence is either always true or always false.

Some examples of open sentences include;

$x+y=10$

$5+n=12$

## Solving open sentences in maths

When you are asked to solve an open sentence you will be given different variables, this is known as a replacement set. You can use these variables to find out whether the open sentence is true or false.

Find the solution for the equation, $3x+6=18$ if the replacement set is {2, 4, 6, 8}

Solution:

To solve this you can start by substituting each option from the replacement set into the equation to find out if any of them create a true sentence;

$3\left(2\right)+6=12$

$\mathbf{3}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{=}\mathbf{18}$

$3\left(6\right)+6=24$

$3\left(8\right)+6=30$

Here you can see that the sentence is true when$x=4$ therefore the solution set is {4}.

Find the solution for the equation, $6\left(x+4\right)=60$ if the replacement set is {2, 4, 6, 8}

Solution:

Once again simply substitute each variable from the replacement set to find your solution;

$6\left(\left(2\right)+4\right)=36$

$6\left(\left(4\right)+4\right)=48$

$\mathbf{6}\mathbf{\left(}\mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{\right)}\mathbf{=}\mathbf{60}$

$6\left(\left(8+4\right)=72$

Here you can see that the sentence is true when$x=6$, therefore your solution set is {6}.

Find the solution for the equation, $12-2x=-4$ if the replacement set is {2, 4, 6, 8}

Solution:

Start by substituting each of the variables into the equation;

$12-2\left(2\right)=8$

$12-2\left(4\right)=4$

$12-2\left(6\right)=0$

$\mathbf{12}\mathbf{-}\mathbf{2}\mathbf{\left(}\mathbf{8}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{4}$

Here you can see that the sentence is true when$x=8$, therefore the solution set is {8).

## What are identities?

An identity in maths can be described as a sum that is always true.

There are different types of identities that have different properties.

When adding any number to 0, the sum will be equal to the number, therefore 0 is the additive identity.

$10+0=10$

### Multiplicative identity

When multiplying a number with 1 the product is equal to the number, therefore 1 is the multiplication identity.

$5×1=5$

### Multiplicative property of zero

When multiplying a number with 0, the product is equal to 0.

$12×0=0$

### Multiplicative inverses

This is when two numbers are multiplied to equal 1, this is also known as reciprocals.

$\frac{1}{5}×5=1$

## Solving identities in maths

When it comes to solving identities you may be asked to solve the missing variable or identify which property of identities is shown.

Find the value of x and identify what property is being used, $25×x=25$

Solution:

You know that the multiplicative identity says that a number is multiplied by 1 the product is the number, therefore $x=1$.

Find the value of x and identify what property is being used, $32+x=32$

Solution:

You know that the additive identity says that when you add 0 to a number, the sum is the number, therefore$x=0$.

Find the value of x and identify what property is being used, $2×x=1$

Solution:

The multiplicative inverse property is applied when the product of two numbers equals 1. This means that$x=\frac{1}{2}$.

## Open Sentences and Identities - Key takeaways

• Open sentences are a type of mathematical sentence that involve variables making it unknown whether it is true or false.
• A replacement set is a list of potential variables to help you solve the open sentence.
• There are 4 different identity properties:
• Multiplicative identity
• Multiplicative property of zero
• Multiplicative inverses, also known as reciprocals

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What are open sentences and identities in maths?

An open sentence in maths is a sentence where it is not known whether it is true or false. However, identities in maths can be described as a sum that is always true .

What is an example of open sentences?

Some examples of open sentences are;

• x + 5 = 12
• y - 6 = 23
• 3 + n = 8

What are the rules for solving open sentences and identities in maths?

When solving open sentences you are often given a replacement set, the variables in this set can be substituted into the sentence in order to determine whether it is true or false.

What are the variables in an open sentence?

A variable in an open sentence represents a number that you do not yet know the value of, it is often represented as an x or y.

What are the rules for solving open sentences in maths?

When solving an open sentence you are finding out the missing variable to check whether the sentence is true or false.

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