Linear inequalities play a crucial role in mathematics and are essential tools for representing relationships between variables. When graphed, linear inequalities form regions in the coordinate plane that represent all possible solutions that satisfy the inequality. In this article, we will explore the graph of the linear inequality 2x – 3y < 12 and discuss how to interpret and work with such inequalities graphically.
Understanding Linear Inequalities
A linear inequality is an inequality that involves linear expressions, which means they can be written in the form of Ax + By < C, where A, B, and C are constants, and x and y are variables. The inequality can be one of several forms: less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥).
In our case, we are interested in the linear inequality 2x – 3y < 12. To understand its graph, we will first manipulate it to find its boundary line and then determine which side of the line satisfies the inequality.
Finding the Boundary Line
To find the boundary line for 2x – 3y < 12, we start by rewriting it as an equation:
2x – 3y = 12.
Now, we can solve this equation for y to put it in slope-intercept form (y = mx + b), where m is the slope, and b is the y-intercept:
-3y = -2x + 12 y = (2/3)x – 4.
So, the equation 2x – 3y = 12 represents a line with a slope of 2/3 and a y-intercept of -4.
Graphing the Boundary Line
Now that we have the boundary line y = (2/3)x – 4, we can graph it on the coordinate plane. To do this, we can plot the y-intercept at (0, -4) and then use the slope to find additional points. For example, if we increase x by 3 (since the slope is 2/3), we get (3, -2), and if we decrease x by 3, we get (-3, -6).
Plotting these points and drawing a straight line through them, we have the graph of the boundary line:
Shading the Solution Region
Now, let’s consider the original inequality, 2x – 3y < 12. To determine which side of the boundary line satisfies the inequality, we can choose a test point not on the line. A convenient test point is the origin (0,0).
Substitute the coordinates of the test point into the inequality:
2(0) – 3(0) < 12 0 < 12.
Since the inequality is true when we substitute (0,0), the region containing the origin (0,0) satisfies the inequality. This means the region below the boundary line is the solution to 2x – 3y < 12.
Shading the region below the boundary line on the graph, we have:
[Insert Shaded Graph]
Interpreting the Solution
The shaded region represents all the points (x, y) that satisfy the inequality 2x – 3y < 12. In other words, any point within this shaded area, when plugged into the inequality, will result in a true statement.
It’s important to note that the boundary line itself is not included in the solution because the inequality is strict (<), not less than or equal to (≤). Points on the boundary line would make the inequality true if it were 2x – 3y ≤ 12.
Understanding and graphing linear inequalities are essential skills in mathematics, particularly in algebra and geometry. In this article, we explored the linear inequality 2x – 3y < 12 and learned how to find its boundary line, graph the boundary line, and shade the region that satisfies the inequality.
By following these steps, we were able to interpret the graph and determine that the solution to 2x – 3y < 12 lies below the boundary line on the coordinate plane. This knowledge can be applied to solve real-world problems where linear inequalities are used to represent various constraints and conditions.
To further understand the practical implications of the graphed linear inequality 2x – 3y < 12, let’s consider a real-world scenario where such inequalities are commonly applied.
Real-World Application: Budgeting and Expenses
Imagine you are managing your monthly budget, and you want to ensure that your expenses stay within a certain limit. In this case, the inequality 2x – 3y < 12 could represent your budget constraint, where:
x represents the amount you spend on essential items (e.g., rent, groceries).
y represents the amount you spend on discretionary or non-essential items (e.g., dining out, entertainment).
The constant 12 represents your total monthly budget.
The graphed inequality visually illustrates the financial boundaries you must adhere to in order to stay within your budget. Here’s how you can interpret the graph in this context:
Region Below the Line: The shaded region below the boundary line represents all the combinations of essential and non-essential expenses (x and y values) that keep you within your budget. Any point within this region corresponds to a budget-friendly combination of spending.
Points on the Line: The points lying exactly on the boundary line (2x – 3y = 12) represent scenarios where you spend your entire budget, with no room for additional expenses. In other words, your spending equals your budget.
Region Above the Line: Points above the boundary line represent combinations of expenses that exceed your budget. These scenarios would result in overspending, which you aim to avoid.
By referring to the graph and considering your actual expenses for the month, you can easily determine whether you are staying within your budget. For example, if your essential expenses (x) are $500, and your non-essential expenses (y) are $200, you can plot the point (500, 200) on the graph. If this point falls within the shaded region below the line, it means you are within budget. However, if the point falls above the line, it indicates overspending.
This simple example demonstrates how linear inequalities, such as 2x – 3y < 12, can be applied to real-life situations to manage finances, allocate resources efficiently, and make informed decisions based on constraints.
In conclusion, understanding the graph of a linear inequality like 2x – 3y < 12 is not only a valuable mathematical skill but also a practical one that can be used in various real-world scenarios. Whether you’re managing a budget, optimizing resources, or solving complex problems with constraints, the ability to interpret and work with linear inequalities is an essential tool for making informed decisions and achieving your goals.