Question Video: Optimization Using Inequalities Mathematics

Video Transcript

Given that 𝑥 is greater than or equal to negative six and less than or equal to 14 and 𝑦 is greater than or equal to eight and less than or equal to 14, find the smallest possible value of 𝑥𝑦.

In this question, we are trying to find the smallest possible value of the product of our two variables. We recall that when we multiply a positive number by a negative number, we get a negative answer. This is true no matter which order the two numbers are written. This means that in order to achieve the smallest possible product, we need one of the variables to be positive and one to be negative. We see that 𝑦 will always be positive as it must be greater than or equal to eight and less than or equal to 14. The variable 𝑥, on the other hand, can take positive or negative values. It is greater than or equal to negative six and less than or equal to 14.

The number furthest away from zero or most negative that 𝑥 can take is negative six. This means that the smallest or most negative possible value of 𝑥𝑦 will occur when 𝑥 is equal to negative six. For the product to be as small as possible, we require 𝑦 to be as large as possible. Therefore, 𝑦 is equal to 14. The smallest possible value of 𝑥𝑦 is therefore equal to negative six multiplied by 14. We know that six multiplied by 10 is equal to 60, and six multiplied by four is equal to 24. This means that six multiplied by 14 is equal to the sum of these two values, which is 84. Negative six multiplied by 14 is therefore equal to negative 84. And this is the smallest possible value of 𝑥𝑦.