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**Hint:**Quantifiers are special phrases in mathematics. This lesson defines quantifiers and explores the different types in mathematical logic.

In mathematical logic, there are two quantifiers: 'there exists' and 'for all.'

There is a certain symbol for these two quantifiers.

Phrase: “there exists”

Symbol: \[\exists \]

Phrase: “for all”

Symbol: \[\forall \]

Using the symbols we can find the suitable quantifier.

**Complete step-by-step answer:**

We have to find the suitable quantifier for \[x + 1 > x\] for all real values of \[x\].

We know that; quantifiers are special phrases in mathematics. This lesson defines quantifiers and explores the different types in mathematical logic.

Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: 'there exists' and 'for all.'

The phrase 'there exists' is called an existential quantifier, which indicates that at least one element exists that satisfies a certain property.

The phrase 'for all' is called a universal quantifier, and it indicates that all the elements of a given set satisfy a property.

There is a certain symbol for these two quantifiers.

Phrase: “there exists”

Symbol: \[\exists \]

Phrase: “for all”

Symbol: \[\forall \]

We know that, \[x + 1\] is always greater than \[x\].

Therefore, \[x + 1 > x\] is true for any real values of \[x\].

Symbolically we can write that, \[x + 1 > x\]\[\forall \]\[x\].

Hence, the suitable quantifier for \[x + 1 > x\] for all real values of \[x\] is \[x + 1 > x\]\[\forall \]\[x\].

Thus, the quantifier to describe the variable of the given sentence is Universal quantifier (\[\forall \]).

**Note:**A formula that contains variables is not simply true or false unless each of these variables is bound by a quantifier. If a variable is not bound the truth of the formula is contingent on the value assigned to the variable from the universe of discourse.

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