Linear Equations: how to deduce this inequality is true?

[yucheng]

Homework Statement

The number of positive integers x satisfying the equation

Relevant Equations

$$\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2}=\frac{13}{12}$$

The solution from my book:

From $$\frac{3}{x+2}<\frac{13}{12}<\frac{3}{x} \tag1$$

It follows that $13x<36<13(x+2)$

x<3, i.e. x = 1 or 2. By checking, x=1 is not the solution and x = 2 satisfies the equation.

However, how does the author deduce (1)?

 

[Mark44]

Homework Statement:: The number of positive integers x satisfying the equation
Relevant Equations:: $$\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2}=\frac{13}{12}$$

The solution from my book:

From $$\frac{3}{x+2}<\frac{13}{12}<\frac{3}{x} \tag1$$

It follows that $13x<36<13(x+2)$

x<3, i.e. x = 1 or 2. By checking, x=1 is not the solution and x = 2 satisfies the equation.

However, how does the author deduce (1)?

Of the three numbers, the largest is $\frac 1 x$ and the smallest is $\frac 1{x + 2}$. Three times the largest will be larger than 13/12 and three times the smallest will be less than 13/12.