Graphing Systems Of Inequalities How To Interpret The Graph

Hey guys! So, you're diving into the world of graphing systems of inequalities, and you've got a question about how a specific graph should look, right? No sweat! Let's break it down. We're going to tackle the system:

$\begin{array}{l} y \geq -x - 1 \\ y < \frac{1}{3}x + 3 \end{array}$

and figure out exactly what the graph should look like. This involves understanding the different components of inequalities, how they translate to lines on a graph, and which areas to shade. By the end of this article, you'll not only know how to graph this particular system but also feel confident tackling similar problems.

Understanding Linear Inequalities

Before we jump into the specific system, let's quickly review the basics of linear inequalities. A linear inequality is similar to a linear equation, but instead of an equals sign (=), we have an inequality sign (>, <, ≥, or ≤). These signs tell us that the values on one side are either greater than, less than, greater than or equal to, or less than or equal to the values on the other side.

The general form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. When we're dealing with inequalities, this form changes slightly to include our inequality signs, such as y > mx + b, y < mx + b, y ≥ mx + b, or y ≤ mx + b. These seemingly small changes have significant implications for how we graph these relationships.

When graphing linear inequalities, the first step is to treat the inequality as if it were a regular equation. This means we plot the line y = mx + b on the coordinate plane. The slope (m) tells us the steepness and direction of the line, while the y-intercept (b) tells us where the line crosses the y-axis. For example, in the inequality y ≥ -x - 1, we would first graph the line y = -x - 1. This line has a slope of -1 and a y-intercept of -1.

However, the inequality sign adds an extra layer of complexity. It tells us not just about the line itself, but also about the region of the coordinate plane that satisfies the inequality. This is where shading comes in. If the inequality includes “greater than” (>) or “greater than or equal to” (≥), we shade the region above the line. If it includes “less than” (<) or “less than or equal to” (≤), we shade the region below the line. The shading represents all the points (x, y) that make the inequality true.

Another important distinction is the type of line we draw. For inequalities with “greater than or equal to” (≥) or “less than or equal to” (≤), we draw a solid line. This indicates that the points on the line itself are included in the solution set. For inequalities with “greater than” (>) or “less than” (<), we draw a dashed or dotted line. This means that the points on the line are not part of the solution set; they only serve as a boundary.

Understanding these basics is crucial for graphing systems of inequalities. We need to know how to interpret the slope and y-intercept, how to determine the direction of shading, and how to choose between solid and dashed lines. With these tools in our belt, we can confidently tackle the system of inequalities presented in the question.

Analyzing the First Inequality: y ≥ -x - 1

Let's dive into the first inequality: y ≥ -x - 1. To understand how to graph this, we need to break it down into its key components. First, we'll treat it as a linear equation to find the line, and then we'll consider the inequality to determine shading and the type of line.

The equation we'll initially consider is y = -x - 1. This is in the slope-intercept form y = mx + b, which makes it easy to identify the slope (m) and the y-intercept (b). In this case, the slope (m) is -1, which means for every one unit we move to the right on the graph, we move one unit down. The y-intercept (b) is -1, indicating that the line crosses the y-axis at the point (0, -1).

Now, let's think about the type of line we need to draw. Since our inequality is y ≥ -x - 1, it includes “greater than or equal to.” This means we'll draw a solid line. A solid line tells us that the points on the line itself are part of the solution set. If we had a strict inequality (either > or <), we would use a dashed line to show that the points on the line are not included.

Next up is the shading. The inequality y ≥ -x - 1 tells us that we're interested in the region where y is greater than or equal to -x - 1. On a coordinate plane, “greater than” typically means we shade the region above the line. To confirm this, we can pick a test point that's not on the line, such as (0, 0), and plug it into the inequality:

0 ≥ -0 - 1 0 ≥ -1

This is true! Since the test point (0, 0) satisfies the inequality and it's located above the line, we know we need to shade the region above the line. So, for the inequality y ≥ -x - 1, we're graphing a solid line with a slope of -1 and a y-intercept of -1, and we're shading the region above the line.

Remember, accurately determining the slope, y-intercept, line type (solid or dashed), and shading direction is crucial for correctly graphing inequalities. With this understanding of y ≥ -x - 1, we’re well-equipped to tackle the second inequality and then combine them to find the solution to the system.

Analyzing the Second Inequality: y < (1/3)x + 3

Now, let’s tackle the second inequality in our system: y < (1/3)x + 3. Just like with the first inequality, we'll break it down to understand the line, the type of line, and the shading.

We start by treating the inequality as an equation: y = (1/3)x + 3. This equation is also in slope-intercept form (y = mx + b). Here, the slope (m) is 1/3, which means for every three units we move to the right on the graph, we move one unit up. The y-intercept (b) is 3, so the line crosses the y-axis at the point (0, 3).

Now, let's figure out the type of line. Our inequality is y < (1/3)x + 3, which means “less than.” This tells us that we need to draw a dashed line. Remember, a dashed line indicates that the points on the line itself are not part of the solution set. This is because the inequality is strictly “less than,” not “less than or equal to.”

Next, we need to determine the shading. The inequality y < (1/3)x + 3 tells us we’re interested in the region where y is less than (1/3)x + 3. “Less than” typically means we shade the region below the line. To confirm, let’s use a test point again. The point (0, 0) is a good choice since it’s not on the line. Plugging it into the inequality:

0 < (1/3)(0) + 3 0 < 3

This is true! The test point (0, 0) satisfies the inequality and is located below the line, so we shade the region below the dashed line.

So, for the inequality y < (1/3)x + 3, we’re graphing a dashed line with a slope of 1/3 and a y-intercept of 3, and we’re shading the region below the line. Understanding these components helps us visualize this inequality on the coordinate plane. Now that we've analyzed both inequalities individually, we're ready to combine them to find the solution to the system.

Combining the Inequalities: Finding the Solution

Okay, guys, we've dissected each inequality separately. Now comes the exciting part: combining them to find the solution to the system. The solution to a system of inequalities is the region where the shaded areas of all inequalities overlap. This overlapping region represents all the points that satisfy all the inequalities in the system.

We have two inequalities:

1. y ≥ -x - 1 (Solid line, slope -1, y-intercept -1, shaded above)
2. y < (1/3)x + 3 (Dashed line, slope 1/3, y-intercept 3, shaded below)

Imagine plotting both of these inequalities on the same coordinate plane. You'd have a solid line sloping downwards and a dashed line sloping upwards. Each line has its own shaded region. The solution to the system is where those shaded regions overlap.

Specifically, the inequality y ≥ -x - 1 tells us we're including the solid line itself and the region above it. The inequality y < (1/3)x + 3 tells us we're not including the dashed line, but we are including the region below it.

The overlapping region will be the area that is simultaneously above the solid line and below the dashed line. This region is the solution set for the system of inequalities. It's like a Venn diagram, where the overlapping section is the intersection of the solutions for each inequality.

To visualize this even better, think about how the lines intersect. The point of intersection is crucial because it marks the boundary where the solutions change. To find the exact point of intersection, you would typically solve the system of equations (treating the inequalities as equations): y = -x - 1 and y = (1/3)x + 3. Setting the two expressions for y equal to each other:

-x - 1 = (1/3)x + 3

Solving for x gives us:

-(4/3)x = 4 x = -3

Substituting x = -3 into either equation gives us y = 2. So, the lines intersect at the point (-3, 2). This point is the vertex of the solution region.

Knowing the point of intersection and the shading for each inequality, we can confidently describe the solution region. It’s the area bounded by the solid line y = -x - 1 (included) and the dashed line y = (1/3)x + 3 (not included), where the shading is above the solid line and below the dashed line.

Describing the Graph Correctly

Now, let’s circle back to the original question: “When graphing the following system of inequalities, which of the following statements correctly describes how the graph should look?” We've done the hard work of analyzing each inequality and combining them, so we're in a great position to answer this.

We know the graph will have two lines:

• A solid line represented by y ≥ -x - 1, with a slope of -1 and a y-intercept of -1, shaded above.
• A dashed line represented by y < (1/3)x + 3, with a slope of 1/3 and a y-intercept of 3, shaded below.

The solution region is the area where the shading overlaps – the region above the solid line and below the dashed line. The lines intersect at the point (-3, 2).

With this detailed description, we can evaluate any statements about the graph and choose the one that accurately reflects these characteristics. For instance, let's consider the statement:

“A solid line shaded above the line with a slope of 1…”

This statement is incorrect because our solid line has a slope of -1, not 1. We need to look for a statement that correctly identifies the slopes, line types, shading directions, and any key points like the intersection.

A correct statement might look something like this:

“A graph with a solid line (y ≥ -x - 1) shaded above and a dashed line (y < (1/3)x + 3) shaded below. The solid line has a slope of -1 and a y-intercept of -1, while the dashed line has a slope of 1/3 and a y-intercept of 3. The solution is the region where the shadings overlap.”

This statement captures all the essential details of the graph and the solution to the system of inequalities.

In summary, to accurately describe the graph, you need to consider:

• The type of lines (solid or dashed) for each inequality.
• The slopes and y-intercepts of each line.
• The direction of shading for each inequality.
• The overlapping region that represents the solution to the system.

By paying attention to these details, you can confidently interpret and describe the graphs of systems of inequalities.

Conclusion

Alright, guys, we've taken a comprehensive journey through graphing systems of inequalities! We started with a specific problem and expanded it into a full understanding of the concepts involved. You now know how to analyze linear inequalities, determine the type of line and shading, and combine inequalities to find the solution region. You’re also equipped to describe the graph accurately, paying attention to key details like slopes, y-intercepts, line types, and shading directions.

Graphing systems of inequalities might seem tricky at first, but by breaking it down step by step, you can master this skill. Remember to focus on the individual inequalities, understand their graphical representations, and then combine them to find the solution. Keep practicing, and you'll become a pro in no time! If you ever get stuck, just revisit these steps, and you’ll be back on track.