What is 1/((n-2))+1/((n-1))> 1/n ?

The solution to 1/((n-2))+1/((n-1))> 1/n is -sqrt(2) 2

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1/((n-2))+1/((n-1))> 1/n

\frac{1}{( n - 2 )} + \frac{1}{( n - 1 )} > \frac{1}{n}

- 2 < n < 0 or 1 < n < 2 or n > 2

Interval Notation

(- 2, 0) \cup (1, 2) \cup (2, \infty)

Decimal Notation

- 1.41421 \dots < n < 0 or 1 < n < 1.41421 \dots or n > 2

Solution steps

\frac{1}{n - 2} + \frac{1}{n - 1} > \frac{1}{n}

Rewrite in standard form

\frac{n ^{2} - 2}{n ( n - 2 ) ( n - 1 )} > 0

Factor

\frac{n ^{2} - 2}{n ( n - 2 ) ( n - 1 )} : \frac{( n + 2 ) ( n - 2 )}{n ( n - 2 ) ( n - 1 )}

\frac{( n + 2 ) ( n - 2 )}{n ( n - 2 ) ( n - 1 )} > 0

Identify the intervals

- 2 < n < 0 or 1 < n < 2 or n > 2

(x - 2) (x - 3) (x + 2) (x + 3) > 0

x (9 x - 4) < (3 x + 2)^{2}

(x - 1)^{2021} (x + 1)^{2022} < 0

3 x^{12} > 0

[3 x - 1] < 81

What is 1/((n-2))+1/((n-1))> 1/n ?
The solution to 1/((n-2))+1/((n-1))> 1/n is -sqrt(2)<n<0\lor 1<n<sqrt(2)\lor n>2