Solving Systems Of Inequalities A Step-by-Step Guide To Y + 2x > 3 And Y ≥ 3.5x - 5

In the realm of mathematics, solving systems of inequalities is a fundamental skill with applications spanning various fields, from economics to engineering. This article delves into the process of solving the system of inequalities: y + 2x > 3 and y ≥ 3.5x - 5. We will explore each inequality individually, transforming them into slope-intercept form, identifying their boundary lines, and ultimately determining the solution set that satisfies both inequalities simultaneously. This guide aims to provide a clear, step-by-step approach, making the concepts accessible to learners of all levels.

Understanding the First Inequality: y + 2x > 3

Let's begin by dissecting the first inequality: y + 2x > 3. Our initial goal is to transform this inequality into slope-intercept form. Slope-intercept form is a standard way of representing linear equations, expressed as y = mx + b, where m represents the slope and b represents the y-intercept. This form provides a clear understanding of the line's orientation and position on the coordinate plane. To convert our inequality, we need to isolate y on one side. This involves subtracting 2x from both sides of the inequality. This process maintains the balance of the inequality while moving the 2x term to the right side. The transformed inequality will then reveal the slope and y-intercept, giving us crucial information for graphing. This initial step is critical in visually representing the inequality and understanding its solution set. The slope-intercept form not only simplifies graphing but also allows for a quick interpretation of the line's characteristics, which is essential for determining the region that satisfies the inequality.

After subtracting 2x from both sides, we arrive at y > -2x + 3. This is the slope-intercept form of the first inequality. Now, we can clearly identify the slope and y-intercept. The slope, represented by m, is -2, indicating that the line descends 2 units for every 1 unit increase in the x-direction. This negative slope signifies a downward trend of the line on the graph. The y-intercept, represented by b, is 3, meaning the line intersects the y-axis at the point (0, 3). This point serves as a crucial anchor for graphing the line. The slope and y-intercept together provide a complete picture of the line's orientation and position on the coordinate plane. Understanding these parameters is vital for accurately graphing the boundary line and subsequently identifying the solution region for the inequality. The slope dictates the steepness and direction of the line, while the y-intercept pinpoints where the line crosses the vertical axis, both being fundamental elements in visualizing the linear relationship.

Next, we need to consider the boundary line. Since the inequality is y > -2x + 3, the boundary line is defined by the equation y = -2x + 3. This line acts as a divider on the coordinate plane, separating the region where the inequality holds true from the region where it does not. However, because the inequality uses a strict “greater than” sign (>), the boundary line itself is not included in the solution set. This is a crucial distinction, as it affects how we represent the line on the graph. To indicate that the boundary line is not part of the solution, we draw it as a dashed line. A dashed line visually communicates that the points on the line do not satisfy the inequality, and only the points in the region on one side of the line are valid solutions. In contrast, if the inequality included an “equal to” component (≥ or ≤), we would draw a solid line to indicate that the points on the line are also part of the solution set. Therefore, the dashed line for y = -2x + 3 is a critical visual cue that conveys the exclusivity of the boundary.

Analyzing the Second Inequality: y ≥ 3.5x - 5

Now, let's turn our attention to the second inequality: y ≥ 3.5x - 5. This inequality is already in slope-intercept form, which simplifies our analysis. We can immediately identify the slope and the y-intercept. The slope, in this case, is 3.5, which is a positive value. This indicates that the line rises 3.5 units for every 1 unit increase in the x-direction, signifying an upward trend. A positive slope means that as we move from left to right along the line, the y-values increase. The y-intercept is -5, meaning the line intersects the y-axis at the point (0, -5). This point is where the line crosses the vertical axis, providing another key reference for graphing. Understanding the slope and y-intercept is crucial for accurately representing the line on the coordinate plane and determining the region that satisfies the inequality. The slope gives us the line's direction and steepness, while the y-intercept anchors the line to a specific point on the y-axis, together defining the line's position and orientation.

The boundary line for the second inequality, y ≥ 3.5x - 5, is defined by the equation y = 3.5x - 5. This line separates the coordinate plane into two regions: one where the inequality holds true and another where it does not. Unlike the first inequality, this inequality includes an “equal to” component (≥). This means that the points on the boundary line itself are part of the solution set. This is a critical distinction that affects how we represent the line on the graph. To indicate that the boundary line is included in the solution, we draw it as a solid line. A solid line visually communicates that all the points on the line satisfy the inequality, in addition to the points in the region on one side of the line. This is in contrast to a dashed line, which indicates that the boundary line is not part of the solution. Therefore, the solid line for y = 3.5x - 5 is an important visual cue that signifies the inclusivity of the boundary in the solution set.

Graphing the Inequalities and Finding the Solution Set

To visualize the solution to the system of inequalities, we need to graph both inequalities on the same coordinate plane. First, we graph the line y = -2x + 3 as a dashed line, since the first inequality is y > -2x + 3. We can plot the y-intercept at (0, 3) and use the slope of -2 to find another point on the line, such as (1, 1). Connecting these points with a dashed line gives us the boundary for the first inequality. The dashed line indicates that the points on this line are not part of the solution. Next, we need to determine which side of the line represents the solution region. To do this, we can choose a test point that is not on the line, such as (0, 0), and substitute its coordinates into the inequality. If the inequality holds true for the test point, then the region containing the test point is the solution region. If the inequality does not hold true, then the other region is the solution region. This process of testing a point is crucial for accurately identifying the correct side of the line that satisfies the inequality. By shading the appropriate region, we visually represent all the points that satisfy the condition y > -2x + 3.

For the first inequality, substituting (0, 0) into y > -2x + 3 gives us 0 > -2(0) + 3, which simplifies to 0 > 3. This is false, so the region containing (0, 0) is not the solution region. Therefore, we shade the region above the dashed line y = -2x + 3, as this region represents all the points that satisfy the inequality. Shading the correct region is essential for visually representing the solution set. It allows us to quickly identify which points on the coordinate plane fulfill the inequality condition. The shaded area represents an infinite set of points, each of which, when substituted into the inequality, will result in a true statement. This visual representation makes it easier to understand the scope and nature of the solution.

Next, we graph the line y = 3.5x - 5 as a solid line, since the second inequality is y ≥ 3.5x - 5. We can plot the y-intercept at (0, -5) and use the slope of 3.5 to find another point on the line, such as (2, 2). Connecting these points with a solid line gives us the boundary for the second inequality. The solid line indicates that the points on this line are part of the solution. Similar to the first inequality, we need to determine which side of the line represents the solution region. We can use the same test point method, choosing a point that is not on the line, such as (0, 0), and substitute its coordinates into the inequality. This will help us identify the region that satisfies the condition y ≥ 3.5x - 5.

For the second inequality, substituting (0, 0) into y ≥ 3.5x - 5 gives us 0 ≥ 3.5(0) - 5, which simplifies to 0 ≥ -5. This is true, so the region containing (0, 0) is the solution region. Therefore, we shade the region above the solid line y = 3.5x - 5, as this region represents all the points that satisfy the inequality. The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This overlapping region represents the set of all points (x, y) that satisfy both inequalities simultaneously. It is the intersection of the two solution sets, and any point within this region will make both inequalities true. Visually, the overlapping shaded area provides a clear representation of the combined solution, allowing us to easily identify the points that meet both conditions.

Identifying the Solution Set

The solution set to the system of inequalities y + 2x > 3 and y ≥ 3.5x - 5 is the region where the shaded areas of the two inequalities overlap on the graph. This region represents all the points (x, y) that satisfy both inequalities simultaneously. To accurately identify this region, it's crucial to consider the nature of the boundary lines. The boundary line for y + 2x > 3 is a dashed line, indicating that the points on this line are not part of the solution. In contrast, the boundary line for y ≥ 3.5x - 5 is a solid line, indicating that the points on this line are included in the solution. Therefore, the solution set includes all points in the overlapping shaded region, as well as the points on the solid boundary line, but excludes the points on the dashed boundary line. This distinction is vital for a complete and accurate representation of the solution.

In summary, solving systems of inequalities involves transforming inequalities into slope-intercept form, graphing the boundary lines (using dashed lines for strict inequalities and solid lines for inclusive inequalities), determining the solution regions for each inequality, and identifying the overlapping region that represents the solution set. This process combines algebraic manipulation with graphical representation to provide a comprehensive understanding of the solution. The overlapping region is the key to the solution, as it visually represents the set of all points that satisfy all the inequalities in the system. By understanding these steps, one can confidently tackle a wide range of systems of inequalities, applying these principles to various mathematical and real-world problems.

This comprehensive guide has walked you through the process of solving the system of inequalities y + 2x > 3 and y ≥ 3.5x - 5. By understanding the individual inequalities, their graphical representations, and the concept of overlapping solution regions, you can confidently solve similar problems and apply these skills in various mathematical and practical contexts.