Problem 73 - College Algebra Essentials

Question

Without graphing, 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 system has no solution because there's no overlap between the regions determined by both inequalities.

Step 1: Analyze the First Inequality

The first inequality is \(3x + y < 9\). This can be rearranged to get \(y < 9 - 3x\). It represents a region below the line \(y = 9 - 3x\).

Step 2: Analyze the Second Inequality

The second inequality is \(3x + y > 9\). This can be rearranged to get \(y > 9 - 3x\). It represents a region above the line \(y = 9 - 3x\).

Step 3: Compare the Two Inequalities

Considering both inequalities, it becomes evident that there's no overlap between the two regions determined by the inequalities. The first inequality indicates a region below the line, whereas the second inequality indicates a region above the same line. Therefore, there's no point that would satisfy both inequalities simultaneously.

Key Concepts

Solutions to Inequalities

In mathematics, inequalities are used to describe the relative size or order of two values. When dealing with a system of inequalities, we look for a common solution that satisfies all inequalities in the system. A solution to an inequality is any number that, when substituted into the inequality, makes it a true statement. For example, if we have the inequality \(x < 5\), any number less than 5 would be considered a solution.

When solving a system of inequalities, the solutions are points that satisfy all the inequalities simultaneously. In the case of the original exercise, we have two inequalities: \(3x + y < 9\) and \(3x + y > 9\). Here, a solution would be a point \((x, y)\) that makes both inequalities true.

However, upon analyzing these inequalities, we discover that there's a contradiction, as no single \((x, y)\) pair can simultaneously fall into the regions described by these inequalities.

Graphical Representation of Inequalities

Graphically, inequalities can be represented on a coordinate plane, helping us visualize the set of possible solutions. Each inequality divides the plane into two halves. One region satisfies the inequality, while the other does not. For linear inequalities like \(y < 9 - 3x\) and \(y > 9 - 3x\), the boundary is a straight line represented by the equation \(y = 9 - 3x\).

To graph an inequality, first draw its boundary line. If the inequality is strict (less than or greater than), the line is typically dashed, indicating points on the line are not included in the solution set. If the inequality includes equal (\(\leq\) or \(\geq\)), the line is solid. Then, shade the region that represents the solution set.

In this specific case of the exercise, graphing would show the regions \(y < 9 - 3x\) and \(y > 9 - 3x\) as non-overlapping, meaning there are no points that satisfy both inequalities at once.

Linear Inequalities

Linear inequalities involve variables raised to the first power and can be visualized as half-planes on a graph. They come in different forms such as \(ax + by < c\), where \(a\), \(b\), and \(c\) are constants. The inequality divides the coordinate plane into two distinct regions, separated by a boundary line, which delineates the solutions from the non-solutions.

The process of converting an inequality into a line equation involves expressing one variable in terms of the other. For example, for \(3x + y < 9\), rearranging gives us \(y < 9 - 3x\). This form makes it easier to determine which side of the boundary line an \((x, y)\) pair should be part of.

Understanding the arrangement and implications of linear inequalities is crucial for solving these systems. In very rare cases, as shown in the exercise, both inequalities disallow overlap completely, resulting in no solutions where both conditions can be satisfied.