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  4. The number of pairs of consecutive even positive integers, both of which are larger than 8 , such that their sum is less than 25 , is

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Q. The number of pairs of consecutive even positive integers, both of which are larger than 8 , such that their sum is less than 25 , is

Linear Inequalities

Solution:

Let $x$ be the smaller of two consecutive even positive integers. Then, the other even integer is $x+2$.
$ \text { Given, } x>8 \text { and } x+x+2<25 $
$ \Rightarrow x>8 \text { and } 2 x+2<25 $
$ \Rightarrow x>8 \text { and } 2 x<23$
$ \Rightarrow x>8 \text { and } x<\frac{23}{2} $
$ \Rightarrow x=10$
Hence, there exists only one pair of even integer $(10,12)$.