Problem 66 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The given system of inequalities has infinitely many solutions.

Step 1: Analysis of the Inequalities

Identify the inequalities in the system. They are: \(3x + y \leq 9\) and \(3x + y \geq 9\).

Step 2: Determine the nature of system

Check if the inequalities are in contradiction or coincide. Because the left sides of both inequalities are identical (3x + y), while the right sides are also identical (9), and the direction of the inequalities allow a solution in both cases (less than or equals to 9, and greater than or equals to 9), we can infer that both equations are the same and therefore coincide.

Step 3: Conclusion

When both equations coincide, it means they represent the same equation and will have all solutions in common. Therefore, this system of inequalities has infinitely many solutions.

Key Concepts

Infinitely Many Solutions

In this system of inequalities, we come across a fascinating scenario: infinitely many solutions. But what does this mean? Simply put, such systems have a limitless number of solutions that satisfy all inequalities at once.

When we analyzed the inequalities, we noticed they are essentially the same: \(3x + y \leq 9\) and \(3x + y \geq 9\). Since these describe the same linear boundary where one is less than or equal to and the other is greater than or equal to the same value, every point on the line \(3x + y = 9\) works. This ensures that there are infinite intersections of the two inequalities, all occurring at every point along this line.

Inequalities

Inequalities are expressions that show the relationship between two values where they are not strictly equal.

In the system of inequalities provided, we see two expressions: \(3x + y \leq 9\) and \(3x + y \geq 9\). These inequalities might seem similar but present key ideas:

• The first inequality (\(3x + y \leq 9\)) suggests that anything on or below the line is part of the solution set.
• The second inequality (\(3x + y \geq 9\)) means solutions are on or above the line.

The beauty is where these overlap—all real numbers meeting both conditions lie along \(3x + y = 9\). This overlap shows that inequalities can define regions that intersect along lines or even specific shaded areas, creating a richer understanding of potential solutions.

Mathematical Analysis

Mathematical analysis involves closely examining mathematical concepts and structures. In this exercise, analysis helped us break down and understand the behavior of given inequalities without graphing.

By considering both sides of the inequalities—\(3x + y \)—and identifying their relationship to the constant (9), we logically deduced that they are identical expressions. Through analysis:

• We confirmed both inequalities describe the same scenario, reinforcing that they have infinitely many solutions.
• This understanding stems from recognizing the equal influence of both inequalities on possible solutions.

Analysis allows us to solve complex problems efficiently by mapping relationships and drawing precise conclusions, crucial in algebraic contexts like this.