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Find a system of inequalities whose solution set is unbounded.

Short Answer

Expert verified
The system of inequalities that has an unbounded solution set is: \[ \begin{cases} x + y > 0 \\ y < x + 4 \end{cases} \] This allows the variables \(x\) and \(y\) to extend infinitely without any restrictions.

Step by step solution

01

Choose the first inequality

We will start by writing down the first inequality. Choose an inequality such that it doesn't restrict the values of both variables in a specific direction. For this purpose, let's start with the inequality: \[x + y > 0\]
02

Choose the second inequality

Let's choose the second inequality to be: \[ y < x + 4\] Combining these two inequalities, our system of inequalities will look like: \[ \begin{cases} x + y > 0 \\ y < x + 4 \end{cases} \]
03

Analyzing the solution set

Now, let's analyze the solution set for this system of inequalities. For the first inequality, \(x + y > 0\), we can see that when \(x\) or \(y\) becomes larger, the inequality still holds true. There is no upper bound for \(x\) or \(y\). As for the second inequality, \(y < x + 4\), we can notice that for any given value of \(x\), we can choose a value of \(y\) that is less than \(x + 4\) (e.g., \(y = x + 3\)). In a similar fashion, there is also no upper bound for \(x\), since the inequality doesn't restrict the values of \(x\) in any specific way. The combination of these inequalities forms a region in the xy-plane where the solution set is unbounded because there are no limitations on the variables in any specific direction.

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