A man wants to cut three lengths from a single piece of board of length 91 cm.

Solution:

Let the length of the shortest piece be x cm. Then, length of the second piece and the third piece are (x + 3) cm and 2x cm respectively.

Since the three lengths are to be cut from a single piece of board of length 91 cm,

$x \mathrm{~cm}+(x+3) \mathrm{cm}+2 x \mathrm{~cm} \leq 91 \mathrm{~cm}$

$\Rightarrow 4 x+3 \leq 91$

$\Rightarrow 4 x \leq 91-3$

$\Rightarrow 4 x \leq 88$

$\Rightarrow \frac{4 x}{4} \leq \frac{88}{4}$

$\Rightarrow x \leq 22$   (1)

Also, the third piece is at least 5 cm longer than the second piece.

$\therefore 2 x \geq(x+3)+5$

$\Rightarrow 2 x \geq x+8$

 

$\Rightarrow x \geq 8 \ldots(2)$

From (1) and (2), we obtain

$8 \leq x \leq 22$

Thus, the possible length of the shortest board is greater than or equal to 8 cm but less than or equal to 22 cm.