Solving System Of Inequalities: A Step-by-Step Guide

Navigating the world of inequalities can sometimes feel like traversing a complex maze, but fear not! In this guide, we'll break down the process of solving a system of inequalities step by step. Specifically, we'll tackle the following system:

$\begin{aligned} 3y &> 2x + 12 \\ 2x + y &\leq -5 \end{aligned}$

By the end of this article, you'll have a clear understanding of how to solve such systems and interpret their solutions graphically.

Understanding Inequalities

Before we dive into solving the system, let's briefly discuss what inequalities are. Unlike equations, which state that two expressions are equal, inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols used to represent these relationships are >, <, ≥, and ≤, respectively.

When dealing with a system of inequalities, we're essentially looking for the region in the coordinate plane that satisfies all the inequalities simultaneously. This region is the intersection of the solution sets of each individual inequality. To find this region, we'll graph each inequality and identify the area where their shaded regions overlap.

Step 1: Isolate 'y' in Each Inequality

To make graphing easier, our first step is to isolate 'y' in each inequality. This puts the inequalities in a slope-intercept-like form, which allows us to quickly identify the slope and y-intercept of the boundary lines.

Inequality 1: 3y > 2x + 12

To isolate 'y', we divide both sides of the inequality by 3:

$y > \frac{2}{3}x + 4$

This tells us that the boundary line has a slope of $\frac{2}{3}$ and a y-intercept of 4. Also, since the inequality is '>', we know that the solution set includes all points above the boundary line, but not the line itself (represented by a dashed line).

Inequality 2: 2x + y ≤ -5

To isolate 'y', we subtract 2x from both sides of the inequality:

$y \leq -2x - 5$

Here, the boundary line has a slope of -2 and a y-intercept of -5. Because the inequality is '≤', the solution set includes all points below the boundary line, including the line itself (represented by a solid line).

Step 2: Graphing the Inequalities

Now that we have 'y' isolated in both inequalities, we can graph them. Let's start with the first inequality:

Graphing y > (2/3)x + 4

1. Draw the boundary line: Draw a dashed line (since it's a strict inequality, '>') with a slope of $\frac{2}{3}$ and a y-intercept of 4. This means for every 3 units you move to the right, you move 2 units up.
2. Shade the region: Since $y$ is greater than the expression, shade the region above the dashed line. This shaded area represents all the points that satisfy the inequality.

Graphing y ≤ -2x - 5

1. Draw the boundary line: Draw a solid line (since it's an inclusive inequality, '≤') with a slope of -2 and a y-intercept of -5. This means for every 1 unit you move to the right, you move 2 units down.
2. Shade the region: Since $y$ is less than or equal to the expression, shade the region below the solid line. This shaded area represents all the points that satisfy the inequality.

Step 3: Identifying the Solution Set

The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This overlapping region represents all the points (x, y) that satisfy both inequalities simultaneously. Carefully observe your graph to identify this region. It is the intersection of the two solution sets.

To further clarify, any point within this overlapping region, when plugged into both original inequalities, will make both inequalities true. Points outside this region will fail to satisfy at least one of the inequalities.

Step 4: Verifying the Solution

To verify our solution, we can pick a test point within the overlapping region and plug it into both original inequalities. If both inequalities hold true, then our solution set is likely correct.

Let's choose the point (-6, 0) as a test point (make sure this point visually appears to be within the overlapping region on your graph).

Testing with Inequality 1: 3y > 2x + 12

Substituting x = -6 and y = 0:

$3(0) > 2(-6) + 12$

$0 > -12 + 12$

$0 > 0$ This is false. Since our chosen point doesn't satisfy the inequality it indicates that our region of overlap identified on the graph is not correct. After a reviewing of the graph and the equations we realize there is no overlap of the two equations. Meaning that this system of equations has no solution.

Common Mistakes to Avoid

• Using the wrong type of line: Remember to use a dashed line for strict inequalities (>, <) and a solid line for inclusive inequalities (≥, ≤).
• Shading the wrong region: Carefully determine which side of the line to shade based on the inequality symbol. If y > ..., shade above the line; if y < ..., shade below the line.
• Forgetting to isolate 'y': Isolating 'y' is crucial for correctly identifying the slope and y-intercept of the boundary lines.
• Not checking the solution: Always verify your solution by picking a test point within the overlapping region and plugging it into the original inequalities.

Applications of Systems of Inequalities

Systems of inequalities aren't just abstract mathematical concepts; they have real-world applications in various fields. For example:

• Linear Programming: Used to optimize solutions in business and economics, such as maximizing profits or minimizing costs, subject to certain constraints.
• Resource Allocation: Determining the best way to allocate resources, such as labor, materials, and equipment, to meet certain goals.
• Constraint Satisfaction: Solving problems where certain conditions or limitations must be met, such as scheduling tasks or designing circuits.

Conclusion

Solving systems of inequalities involves isolating 'y' in each inequality, graphing the boundary lines, shading the appropriate regions, and identifying the overlapping region that represents the solution set. By following these steps and avoiding common mistakes, you can confidently tackle various systems of inequalities. Remember to verify your solution to ensure accuracy. Although in the example we provided has no solution, this is still a valid answer and the process to get to that answer is still accurate.

To deepen your understanding of inequalities and their applications, consider exploring resources like Khan Academy's article on two-variable linear inequalities.