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Chapter 7: Problem 66

Without graphing, Determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}3 x+y \leq 9 \\ 3 x+y \geq 9\end{array}\right.\)

Short Answer

Expert verified
The given system of inequalities will have infinitely many solutions because both inequalities represent the same line in the plane and every solution to one inequality is also a solution to the other.

Step by step solution

01

Analyzing the first inequality

The first inequality is '3x + y 鈮� 9'. In a system of equations, if the method of substitution or elimination is used to equalize two equations, and if on simplification the equation comes out to be a valid mathematical fact, then it would most likely have infinitely many solutions. However, an inequality equation is considered valid鈥攖he equation having an infinite solution鈥攊f the inequality clause is satisfied.
02

Analyzing the second inequality

The second inequality is '3x + y 鈮� 9'. We will apply the same analysis method as done for the first inequality.
03

Evaluating the possibility of having solutions

As both inequalities have the same left side expressions, and the corresponding right side values are identical, it can be stated that every solution of the first inequality is also a solution of the second one, and vice versa. Thus, the system of these two inequalities will have infinitely many solutions. This line is where the two inequalities meet.

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