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Chapter 7: Problem 64

Without graphing, in Exercises 64-67, determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}3 x+y<9 \\ 3 x+y>9\end{array}\right.\)

Short Answer

Expert verified
The given system of inequalities has no solution.

Step by step solution

01

Analyzing the system of inequalities

First, let's look at the given system of inequalities: \\(\begin{cases}3 x+y<9 \\3 x+y>9\end{cases}\\). Here, both inequalities have exactly the same left side i.e. \(3x+y\) but their right side values are the same and the inequality directions are different.
02

Evaluating the possibility of solutions

Next, we need to see if there is any value of \(x\) and \(y\) that can satisfy both inequalities at the same time. Since the right sides of inequalities are same i.e. 9 but the inequality symbols for two equations are in different directions, one being 'less than' and other being 'greater than'. This means that no matter what values we choose for \(x\) and \(y\), one of the inequalities will always contradict the other one. Therefore, these inequalities will not have any common solutions.

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