 Hi and how are you all? I am Priyanka and let us discuss, says a man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if there is another condition given to us if the third piece is to be at least 5 cm longer than the second. Now this question involves two inequalities that we will be discussing one by one. First of all it's saying that everything is related to the shortest length of the board. So let the length of the shortest board be equal to x cm. Then according to the condition or conditions we must say we are given that the length of the second board and length of the third is equal to the second length is to be 3 cm longer than the shortest. The shortest is x and it has to be 3 cm longer. So this fulfills the condition and third length is to be twice as long as the shortest that means twice as long as the shortest. So this completes defining the length of all the three boards. Now the condition given to us is that first of all the sum of all these should be less than equal to 91 cm because the single piece of board is of length 91 cm. Then it's also given to us in the question that the third piece is to be at least 5 cm longer than the second. That means 2x needs to be greater than or equal to the second board's length plus an additional 5 cm. So these are the two inequalities which are given to us in the question. We will be solving them one by one. Now here we have 4x plus 3 is less than equal to 91 or on subtracting 3 from both the sides we have 4x is less than equal to 88. On dividing both the sides by 4 that is a positive number we have x is less than equal to 22. Right now let us solve this inequality we have 2x is greater than equal to x plus 3 plus 5. 2x is greater than equal to x plus 8. On subtracting x from both the sides we have 2x minus x is greater than equal to x plus 8 minus x or x is greater than equal to 8. So these are the two values that we have received. So the length of the shortest board is greater than equal to 8 but it has to be less than equal to 2 also. So it says that the value of x has to be greater than equal to 8 but less than equal to 22. So putting this answer into words we can say that length of the shortest board is greater than equal to equal to 22. So this answers the question that was given to us. I hope you enjoyed the session. Bye for now.