# Question Video: Determining the System of Inequalities That Describes a Given Situation Mathematics

A carpenter wants to buy two types of nails. The first type cost 6 pounds per kilogram, and the second type cost 9 pounds per kilogram. He needs at least 5 kg of the first type and at least 7 kg of the second. He can spend less than 55 pounds. Using π₯ to represent the amount of the first type and π¦ to represent the second type, state the system of inequalities that represents this situation.

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### Video Transcript

A carpenter wants to buy two types of nails. The first type cost six pounds per kilogram, and the second type cost nine pounds per kilogram. He needs at least five kilograms of the first type and at least seven kilograms of the second. He can spend less than 55 pounds. Using π₯ to represent the amount of the first type and π¦ to represent the second type, state the system of inequalities that represents this situation.

The first type cost six pounds per kilogram. And we need at least five kilograms of the first type. π₯ is our variable for type one. And π¦ is our variable for type two. In the second type, we need at least seven kilograms. Our variables represent the amount of each type we purchased. And that means that our π₯-measure is a value in kilograms. Itβs saying this is how many kilograms of the first type we purchased. And our π¦-value would also be a representation of kilograms.

We need at least five kilograms of the first type. At least means five or more. So our π₯-value needs to be greater than or equal to five. And for the second type, we need at least seven kilograms. At least means greater than or equal to. And weβre dealing with seven. But what would be the inequality that represents the price? We know that the cost of π₯ plus the cost of π¦ has to be less than 55. That means it cannot equal 55. It must be smaller than 55.

The cost of π₯ will be the price of the first type times the amount that the carpenter purchases. The first set of nails cost six pounds per kilogram. The amount that we purchased in kilograms of the first type is π₯. The cost of the second type will be the price of the second type times the amount in kilograms that the carpenter purchases. The price of the second type in kilograms is nine pounds. And the amount of kilograms we purchased of the second type is our variable π¦.

That means six π₯ plus nine π¦ must be less than 55. π₯ must be greater than or equal to five. π¦ must be greater than or equal to seven. And six π₯ plus nine π¦ must be less than 55.