Problem 73 - Algebra and Trigonometry

Question

Without graphing, in Exercises \(73-76,\) determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} {3 x+y<9} \\ {3 x+y>9} \end{array}\right.$$

Step-by-Step Solution

Answer

The given system of inequalities has no solution.

Step 1: Understanding the given relations

Given inequalities are \(3x + y < 9\) and \(3x + y > 9\). These two inequalities represent two half-spaces divided by two lines whose equations are \(3x + y = 9\). One half-space will contain solutions for the equation where \(3x + y > 9\) and the other half-space will contain solutions for the equation where \(3x + y < 9\).

Step 2: Determine if there is any overlapping region

Since one line is less than and the other is greater than, there are no points (x,y) that can satisfy both inequalities at the same time, since they describe two separate distinct sets of points (or half-spaces) with no overlap.

Step 3: Conclusion

Since there is no overlapping region, the system of inequalities has no solutions. This is because, for a system to have solutions, the same point or points must satisfy all the inequalities simultaneously. In this case, there are no such points, thus the system has no solution.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values where they may not necessarily be equal. They use symbols like "<" (less than), ">" (greater than), "≤" (less than or equal to), and "≥" (greater than or equal to) to describe this relationship.
Inequalities are not the same as equations, as they provide a range of possible values instead of just one. This makes them particularly useful in situations where we are looking for possible solutions rather than a single answer.
For instance, in the given exercise, we have two inequalities:

• The first is: \( 3x + y < 9 \)
• The second is: \( 3x + y > 9 \)

These represent conditions for different regions on a graph. The essence of solving such inequalities lies in finding the regions that satisfy the conditions imposed by each inequality, and understanding how these regions interact.

Interpreting Half-Spaces

In the context of inequalities, half-spaces represent the regions of the coordinate plane that satisfy a linear inequality. Each linear inequality divides the plane into two parts or half-spaces.
For the inequality \( 3x + y < 9 \), the half-space includes all points (x, y) that lie below the line \( 3x + y = 9 \). Conversely, the inequality \( 3x + y > 9 \) includes all points above this line.
The line itself \( 3x + y = 9 \) is the boundary between these two half-spaces. Each half-space represents a range of x and y values that do not include the boundary line, as the inequalities \( 3x + y < 9 \) and \( 3x + y > 9 \) are strict and do not include the line where \( 3x + y = 9 \). Understanding half-spaces helps in visualizing the areas of the graph where solutions to the inequalities may lie.

Algebraic Solutions and Systems of Inequalities

When dealing with systems of inequalities, the goal is often to find a common set of solutions that satisfy all given inequalities simultaneously. This involves understanding each inequality's algebraic representation and, if possible, identifying regions where these overlap.
Algebraic solutions for inequalities typically consist of expressing the solution in terms of intervals or regions that meet the necessary conditions. However, in some cases, like the one in the exercise, no algebraic solution is possible because it is impossible for a single point to meet the criteria set by both inequalities simultaneously.
In our problem, we attempt to find solutions for both \( 3x + y < 9 \) and \( 3x + y > 9 \); however, no single point can satisfy both inequalities at the same time because they describe opposite conditions.

Exploring Overlapping Regions

Overlapping regions are key when solving systems of inequalities. They represent the set of all points that simultaneously satisfy all inequalities in the system. In graphical terms, these are the areas where different half-spaces intersect.
If overlapping regions exist in a system of inequalities, it means there are potential solutions that meet all conditions. Conversely, if there are no overlapping regions, as in this case, then no common solution exists.
In our exercise, since one inequality represents a region where \( 3x + y < 9 \) and the other where \( 3x + y > 9 \), there are no overlapping regions. Each inequality describes a half-space separated by the line \( 3x + y = 9 \), with no point existing where both conditions are true. Thus, the conclusion is that the system has no solution.