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How many solutions does a linear inequality in two variables have?

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: From the question given, we have been asked to find the number of solutions of a linear inequality in two variables. We will assume a linear inequality in two variables and find its solutions and analyse them.

Complete step by step answer:
First of all, we have to learn about the linear inequality in two variables.
Linear inequality in two variables:
For example: If apples cost \[40\]rupees per kilo and pears cost \[53\] rupees per kilo, what combination of apples and pears can I buy with at most \[1000\] rupees?
If I buy \[a\] kilos of apples and \[p\] kilos of pears then I spend \[40a\] apples and \[53p\] on pears. So, there will be \[40a+53p\] in total. So, whatever combination I buy, it must satisfy
\[40a+53p\le 1000\]
This is an example of linear inequality in two variables.
Solutions of a linear inequality in two variables-
Therefore, the solution of a linear inequality in two variables like \[Ax+By > c\] is an ordered pair \[\left( x,y \right)\] that produces a true statement when the values of \[x\] and \[y\] are substituted into the inequality.
We can clearly see that the above equation in the example we have discussed, can have many ordered pairs as a solution for the equation.
Therefore, the solution set of a single linear inequality is always a half plane, so there are infinitely many solutions.
So, for a linear inequality in two variables, there are infinitely many numbers of solutions.

Note: We should be very well aware about the concept of linear inequality in two variables. In this type of question, we should have to take an example and using that example, we can find the answer for the given question. Also, we should be very well known about finding the solutions of a linear inequality in two variables. Similarly we can assume any other inequality and analyse it.
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