Problem 41 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x+y>4} \\ {x+y<-1} \end{array}\right. $$

Step-by-Step Solution

Answer

The system of inequalities has no solution, because there's no subset of the coordinate plane that satisfies both inequalities simultaneously.

Step 1: Solve Each Inequality for y

First, we must express each inequality in the form y > f(x) or y < f(x). So, subtract x from both sides to get y > -x + 4 and y < -x - 1.

Step 2: Graph The Inequalities

Plot the lines y = -x + 4 and y = -x - 1 on the same graph. The first line has y-intercept 4 and slope -1 while the second line has y-intercept -1 and also slope -1. Then, shade the area above the line for first inequality (y > -x + 4), and below the line for the second inequality (y < -x - 1).

Step 3: Identify the Solution Set

Check for a common region that satisfies both inequalities. However, as was initially suspected, since one inequality is demanding the sum of x and y to be greater than 4 and the other to be less than -1, there is no region in the graph that can satisfy both inequalities simultaneously. Hence, there is no solution.

Key Concepts

Graphing Inequalities

When graphing inequalities, we are essentially visualizing the possible solution sets that satisfy a given inequality condition. The first step is to manipulate each inequality into a standard form, such as the ones familiar to us like \( y > mx + b \) for a linear inequality. This allows us to draw the boundary line on a graph.

Consider the line formed from the boundary equation and choose a convenient method to graph it:

• Find the y-intercept, which is where the line crosses the y-axis.
• Calculate the slope, which indicates the direction and steepness of the line.

Once the line is plotted, the next step is to determine which side of the line should be shaded. This will represent the set of possible solutions:

• If the inequality is \( y > mx + b \), you shade above the line.
• If it is \( y < mx + b \), shade below the line.

Make sure to graph all inequalities in the system on the same plane so you can easily analyze overlaps and solution regions.

Linear Inequalities

Linear inequalities are basic algebraic expressions that compare linear equations, often denoted by using inequality symbols like \( >, <, \geq, \) or \( \leq \). These inequalities differ from linear equations because they do not have a single solution but rather a range of solutions that can often be visualized as a region in a coordinate plane.

Steps to handle linear inequalities involve:

• Rearranging the inequality into the standard form \( y \) as a function of \( x \).
• Identifying key elements like slope \( m \) and intercept \( b \) to graph the boundary line.
• Using test points to verify which area provides solutions that satisfy the inequality condition.

A critical aspect of linear inequalities is checking intersection effects when they are placed in a system. Here, inconsistencies in overlapping solution regions can lead to no intersection, just as no area satisfies both \( x + y > 4 \) and \( x + y < -1 \) simultaneously in our exercise.

Solution Set Analysis

Analysis of solution sets in systems of inequalities involves identifying which regions of a graphed plane satisfy all the given inequalities simultaneously. This often involves thorough checking if and where shadings overlap, but it’s crucial for a system like our example where different inequalities might contradict each other's demands.

Consider these points in solution set analysis:

• Intersection of shaded regions indicates a solution set where all inequalities hold true simultaneously.
• Lack of overlapping shaded areas, as seen with \( x + y > 4 \) and \( x + y < -1 \), means there is no feasible set of solutions that satisfy every inequality.

This kind of analysis can reveal whether the system is feasible and where it might break down due to conflicting conditions. It's this final step that confirms, as in this instance, if there’s no solution possible by indicating that the graph does indeed lack any overlapping solution region.