Problem 33 - Intermediate Algebra

Question

How do you know by inspection, without graphing, that the solution set of the system \(\left(\begin{array}{l}3 x-2 y>5 \\ 3 x-2 y<2\end{array}\right)\) is the null set?

Step-by-Step Solution

Answer

The solution set is empty due to the contradictory nature of the inequalities.

Step 1: Understand the Inequalities

We are given the system of inequalities: \(3x - 2y > 5\) and \(3x - 2y < 2\). These inequalities share the same left-hand expression \(3x - 2y\) but have different right-hand values.

Step 2: Analyze the Inequalities

Let's consider what both inequalities mean: \(3x - 2y > 5\) demands that \(3x - 2y\) be strictly greater than 5. Simultaneously, \(3x - 2y < 2\) requires that the same expression be strictly less than 2.

Step 3: Check for Contradictions

Check if it's possible for one expression \(3x - 2y\) to satisfy both conditions. For a real number to be greater than 5, it cannot be at the same time less than 2. This creates a contradiction.

Step 4: Conclude the Solution Set

Since it is impossible for any number to simultaneously satisfy both conditions, there are no values \(x\) and \(y\) that can make both inequalities true at the same time. Thus, the solution set for the system is empty.

Key Concepts

Solution Set

In mathematics, a solution set is a collection of all possible solutions that satisfy a particular equation or system of equations. For a problem involving inequalities, like in our system, we're looking for all pairs

• \((x, y)\) that satisfy both conditions simultaneously. In simpler terms, the solution set is the range of values for \(x\) and \(y\) that solve both inequalities together.
• If even one pair \((x, y)\) cannot satisfy both conditions, then they cannot be in the solution set.

In the exercise given, we have a situation where two inequalities

• \(3x - 2y > 5\)
• \(3x - 2y < 2\)

are simultaneously demanding incompatible outcomes for \(3x - 2y\). This contradiction results in an empty solution set, because there is no possible number that is both greater than 5 and less than 2 at the same time. Thus, we conclude the solution set is null, indicating no solutions exist.

Contradiction in Mathematics

A contradiction in mathematics occurs when two or more statements or conditions cannot be true at the same time. Contradictions often indicate an error in assumptions or in this case, highlight the absence of a solution.
For instance, the inequalities \(3x - 2y > 5\) and \(3x - 2y < 2\) present a contradiction because they impose opposite requirements on the same mathematical expression. If considered separately:

• \(3x - 2y > 5\) means the expression must be greater than 5.
• At the same time, \(3x - 2y < 2\) demands it be less than 2.

As a basic principle of numbers, no value can be higher than 5 while also being below 2. This firmly establishes that a contradiction exists, and no values of \((x, y)\) can satisfy both inequalities.
Thus, when you encounter such contradictions, it often leads directly to concluding that there is no solution.

Inequality Analysis

Inequality analysis involves examining mathematical statements where expressions are not equal, and understanding their implications. When analyzing a system of inequalities, steps typically include:

• Looking at each inequality individually
• Attempting to find a region where both inequalities are true simultaneously

In this exercise, we have the inequalities \(3x - 2y > 5\) and \(3x - 2y < 2\). By analyzing these:

• The inequality \(3x - 2y > 5\) suggests any pairs \((x, y)\) must result in a calculation greater than 5.
• Conversely, \(3x - 2y < 2\) requires these pairs to yield a result less than 2.

No value can simultaneously be both above 5 and below 2.
This realization during inequality analysis leads directly to identifying a contradiction, following which one can determine that the solution set is empty.
Understanding inequality analysis helps you deduce exactly when results are possible, and when contradictions make solutions impossible.