Graphing Systems Of Inequalities: Y ext{The } ext{slope Of The Line} ext{, Which Is 2, Means That For Every 1-unit Increase In X, Y Increases By 2 Units. The Y-intercept Is -3, Meaning The Line Crosses The Y-axis At The Point (0, -3). The Inequality Symbol extgreater extequal ext{ Indicates That The Line Itself Is Solid And The Region Above The Line Is Shaded, Representing All Solutions To The Inequality.} } }

Unpacking the Basics: What are Linear Inequalities?

When we talk about linear inequalities, we're diving into a super practical and visually engaging part of mathematics. Unlike standard equations that give us a single line or point, inequalities open up entire regions on a graph, showing us a whole set of possible solutions. Specifically, we're going to explore the graph of two interesting inequalities: $y \leq 3x + 1$ and $y \geq -x + 2$. Understanding these two is like unlocking a secret map to where all the true solutions lie. Imagine you're trying to figure out all the possible combinations of prices and quantities that keep you under budget \u2013 that's often an inequality problem! These aren't just abstract symbols; they represent real-world constraints and possibilities. For instance, in business, you might use inequalities to model production capacities, resource limitations, or profit targets. Each inequality represents a boundary, and combined, they form a "system" that defines a specific region where all conditions are met. This region is the treasure map we're after, showing us where our solutions truly exist. We'll break down each part step-by-step, starting with how to interpret the inequality symbols, how they determine the type of boundary line (solid or dashed), and perhaps most importantly, which side of the line to "shade" to indicate the solution area. By the end, you'll have a clear picture of how these two inequalities interact and what statements about their graph are actually true. It\u2019s a bit like being a detective, gathering clues from each inequality to solve the bigger mystery of the system as a whole. Think of it as painting a picture, where each stroke (or inequality) adds another layer of detail to our overall understanding of the solution space. We're not just looking for an answer, but rather all the possible answers that satisfy both conditions simultaneously. This is the beauty and utility of working with systems of linear inequalities. They allow us to visualize complex relationships and constraints in a simple, intuitive way on a Cartesian plane. We'll need to pay close attention to the direction of the inequality signs, as they dictate not only the shading but also whether the boundary line itself is included in the solution set. A less than or equal to (\u2264) or greater than or equal to (\u2265) sign means a solid line, including points on the boundary. A less than (<) or greater than (>) sign, however, means a dashed line, indicating that points on the boundary are not part of the solution. This fundamental distinction is crucial for accurately representing the solution set and will be a key part of our analysis for the given system. So, let\u2019s grab our graphing tools and get ready to draw some insightful conclusions about these mathematical expressions and uncover the truth behind the statements presented!

Decoding the Boundary Lines: Slope and Type

Graphing $y \leq 3x + 1$: The First Boundary

Let's kick things off by dissecting our first inequality: $y \leq 3x + 1$. The very first step in graphing any linear inequality is to pretend, just for a moment, that it's a regular old equation. So, we'll focus on $y = 3x + 1$. This familiar form, y = mx + b, immediately gives us two crucial pieces of information: the slope and the y-intercept. Here, our slope (m) is 3, and our y-intercept (b) is 1. Remember, the slope tells us how steep the line is and in which direction it's heading \u2013 for a slope of 3, it means "rise 3, run 1." The y-intercept tells us where the line crosses the y-axis, which is at the point (0, 1). This is our starting point for drawing the line. Now, we need to consider the inequality sign itself: \u2264 (less than or equal to). This little symbol is super important because it dictates two things. First, because it includes "or equal to," it means that all the points on the line $y = 3x + 1$ are actually part of the solution set for this specific inequality. Graphically, this translates to drawing a solid line. If it were just '<' or '>', we'd use a dashed line to show that points on the boundary itself aren't included. So, to graph this, you'd mark (0,1), then from there, go up 3 units and right 1 unit to find another point, say (1,4). Connect these points with a ruler, making sure it's a solid line. This confirms one of our initial statements from the prompt: one boundary line has a slope of 2. Well, we just found that the slope of this line is 3, not 2. So, we're already seeing that one of the initial statements is incorrect based on this first inequality alone. But don't worry, we'll verify the slope of the second line soon enough to be absolutely sure. For now, firmly establish this solid line, as it forms a critical boundary for our overall solution region. This line acts like a fence, separating the solutions from the non-solutions for this inequality, with the fence itself being part of the allowed region. The slope is a measure of the steepness of the line, indicating the rate of change of y with respect to x. A positive slope, like our 3 here, means the line goes upward from left to right. Understanding these fundamental properties is key to accurately graphing and interpreting linear inequalities.

Graphing $y \geq -x + 2$: The Second Boundary

Alright, let's turn our attention to the second inequality in our system: $y \geq -x + 2$. Just like before, our first move is to temporarily treat it as an equation: $y = -x + 2$. This gives us its core characteristics. Looking at the y = mx + b form, we can easily spot the slope (m) and the y-intercept (b). For this line, the slope is -1 (because -x is the same as -1x), and the y-intercept is 2. This means our line crosses the y-axis at the point (0, 2). From there, a slope of -1 means "rise -1, run 1" or "down 1, right 1." So, from (0,2), you'd go down 1 unit and right 1 unit to reach (1,1). Connect these points. Now, let's revisit the inequality symbol: \u2265 (greater than or equal to). Similar to the first inequality, the presence of "or equal to" is a huge hint. It tells us that all the points on the line $y = -x + 2$ are absolutely included in the set of solutions for this particular inequality. Therefore, just like with our first line, this boundary line will also be drawn as a solid line. This is another crucial piece of information for evaluating the initial statements. With both inequalities, $y \leq 3x + 1$ and $y \geq -x + 2$, leading to solid lines, we can already confirm that the statement "Both boundary lines are solid" is indeed true! That's one down! And going back to the statement about the slope, we've now found slopes of 3 and -1. Neither of them is 2, so that statement remains false. It\u2019s essential to be precise with these details because they completely change the graphical representation and, consequently, the accuracy of our conclusions. So, you've now got two solid lines on your graph, crisscrossing somewhere. These lines have effectively divided your coordinate plane into distinct regions, and our next step will be to figure out which of those regions contains the actual solutions to our system of inequalities. A negative slope, like our -1 here, indicates that the line goes downward from left to right. This comprehensive understanding of both the slope and the line type \u2013 solid in this case \u2013 is absolutely fundamental to correctly mapping out the solution space of the entire system.

Shading the Solution: Where Do Our Answers Live?

After drawing our boundary lines, the next big step is to figure out which side of each line to shade. This shading represents all the points that satisfy each individual inequality. When we find the area where the shadings overlap, that's our ultimate solution region for the system.

Shading for $y \leq 3x + 1$

Now that we have our first solid boundary line from $y = 3x + 1$ neatly drawn on the graph, it's time to figure out which side needs to be shaded for the inequality $y \leq 3x + 1$. The \u2264 sign is our guide here. When you see "less than or equal to," it almost always means you need to shade the region below the line. Think about it intuitively: if you pick any point below the line, its y-coordinate will be smaller than the y-coordinate of a point on the line with the same x-value. A fantastic way to confirm this is to use a test point. The easiest test point, if it doesn't fall on your line, is often the origin: (0,0). Let's plug (0,0) into our inequality: Is $0 \leq 3(0) + 1$? Is $0 \leq 0 + 1$? Is $0 \leq 1$? Yes, that's true! Since (0,0) makes the inequality true and (0,0) is below the line $y = 3x + 1$, it confirms that we should shade the entire region below this boundary line. This shading illustrates all the possible (x,y) pairs that satisfy the first condition of our system. It\u2019s like marking all the valid territories on our map for the first rule. So, take your pencil or highlighter and gently shade the area beneath the solid line $y=3x+1$. This visual representation is crucial because it helps us to truly see the infinite number of solutions for this single inequality. It also sets the stage for finding the intersection with the second inequality's shaded region. This process of shading is what brings the inequality to life on the graph, moving beyond just a line to an entire domain of permissible points. This shaded area represents every single coordinate pair that, when plugged into the inequality, makes the statement true. It's not just an arbitrary region; it's a rich collection of potential answers, all adhering to the first rule of our system.

Shading for $y \geq -x + 2$

Next up, let's tackle the shading for our second inequality, $y \geq -x + 2$. We've already drawn its solid boundary line from $y = -x + 2$. Now, the \u2265 sign comes into play. When you encounter "greater than or equal to," it signals that you'll need to shade the region above the line. Again, think about it: for any given x-value, the y-values of points above the line are larger than the y-values of points on the line. Let's use our trusty test point, (0,0), again to verify. Remember, you can only use (0,0) if the line doesn't pass through it, which in this case, $y = -x + 2$ does not pass through (0,0) since $0 \neq -0 + 2$. So, plugging (0,0) into $y \geq -x + 2$: Is $0 \geq -(0) + 2$? Is $0 \geq 0 + 2$? Is $0 \geq 2$? No, that's false! Since (0,0) makes the inequality false and (0,0) is below the line $y = -x + 2$, this confirms that the solution region for this inequality must be the area opposite to where (0,0) lies \u2013 in other words, the region above the line. So, grab your pencil again, perhaps a different color if you're using one, and shade the entire area above the solid line $y=-x+2$. This is where things get exciting because the ultimate solution to our system of inequalities is the area where these two shaded regions overlap. It's where the "rules" of both inequalities are satisfied simultaneously. This brings us to another statement from our prompt: "Both inequalities are shaded below the boundary lines." Based on our shading for $y \geq -x + 2$, which is above its line, we can definitively say that this statement is also false. We're meticulously building our understanding and verifying each point! The overlap of these two shaded regions is the key to identifying the complete solution set for the system. Any point within this overlapping region, including on the solid boundary lines that define it, will satisfy both conditions of the inequalities.

Testing a Point: Is (1,3) a Solution?

With our lines drawn and regions shaded, we're now perfectly positioned to address one of the specific statements from our initial query: "A solution to the system is $(1,3)$." To verify if a specific point is a solution to a system of inequalities, it must satisfy every single inequality in that system. If it fails even one, it's out! It's like needing a perfect score on two different tests to pass the course. Let's take the point (1,3) and rigorously test it against both inequalities.

First, let's plug (1,3) into our first inequality: $y \leq 3x + 1$. Substitute $x=1$ and $y=3$: Is $3 \leq 3(1) + 1$? Is $3 \leq 3 + 1$? Is $3 \leq 4$? Absolutely, yes! This statement is true, meaning the point (1,3) satisfies the first inequality. It lies within or on the boundary of the shaded region for $y \leq 3x + 1$. This is a great start! But remember, a solution to the system needs to satisfy both conditions.

Now, let's test (1,3) in our second inequality: $y \geq -x + 2$. Substitute $x=1$ and $y=3$: Is $3 \geq -(1) + 2$? Is $3 \geq -1 + 2$? Is $3 \geq 1$? Yes, that's true as well! The point (1,3) also satisfies the second inequality. It lies within or on the boundary of the shaded region for $y \geq -x + 2$.

Since the point (1,3) satisfies both inequalities simultaneously, we can confidently conclude that the statement "A solution to the system is $(1,3)$" is indeed true! This is the essence of a solution to a system of inequalities \u2013 it's any point that falls into the overlapping shaded region on your graph, including points on the solid boundary lines themselves where the conditions are met. If you were to look at your graph, you would visually see that the point (1,3) sits right within the cross-hatched or doubly-shaded area that represents the solution set for the entire system. This meticulous testing process is crucial for verifying specific points, bridging the gap between algebraic manipulation and graphical interpretation. It reinforces that the solution to a system isn't just a theoretical concept; it's a tangible region of the plane, and specific points can be confirmed to reside within it through simple substitution. This method provides a powerful way to check our graphical work and ensure our understanding of the solution space is accurate and complete.

Wrapping It Up: What We Learned About These Inequalities

Phew! We've journeyed through the intricacies of graphing linear inequalities, from identifying slopes and intercepts to determining solid lines and shading directions. Now, let's tie everything together and specifically address the initial statements that prompted our exploration of $y \leq 3x + 1$ and $y \geq -x + 2$. This systematic approach allows us to confirm or refute each claim with solid mathematical reasoning and graphical evidence. It's truly satisfying to move from abstract statements to concrete, verifiable facts. We've not only covered the mechanics of graphing but also deeply analyzed what each part of the inequality means for the overall solution.

Let's review each statement one last time:

• "The slope of one boundary line is 2."

  • Through our detailed analysis, we found that the first inequality, $y \leq 3x + 1$, has a slope of 3. The second inequality, $y \geq -x + 2$, has a slope of -1. Neither of these slopes is 2. Therefore, this statement is definitively false. It's a classic distractor, testing your ability to correctly identify the 'm' in the y=mx+b form. This highlights the importance of paying close attention to the coefficients in the linear equations that define the boundary lines.

• "Both boundary lines are solid."

  • We carefully observed the inequality symbols. For $y \leq 3x + 1$, the "less than or equal to" (\u2264) symbol dictates a solid line. Similarly, for $y \geq -x + 2$, the "greater than or equal to" (\u2265) symbol also requires a solid line. Because both inequalities include the "or equal to" part, the points on both boundary lines are part of their respective solution sets. So, we can proudly confirm that this statement is true. This is a vital distinction in graphing inequalities, as a dashed line would imply the boundary itself is excluded from the solution.

• "A solution to the system is $(1,3)$."

  • We rigorously tested the point (1,3) in both inequalities. For $y \leq 3x + 1$, we found $3 \leq 4$, which is true. For $y \geq -x + 2$, we found $3 \geq 1$, which is also true. Since (1,3) satisfies both conditions, it falls squarely within the overlapping shaded region on the graph. Thus, this statement is absolutely true. This reinforces the idea that solutions to a system of inequalities must fulfill all conditions simultaneously, and testing points is a reliable way to confirm this.

• "Both inequalities are shaded below the boundary lines."

  • When we examined $y \leq 3x + 1$, the "less than or equal to" sign indeed told us to shade below its boundary line. However, for $y \geq -x + 2$, the "greater than or equal to" sign instructed us to shade above its boundary line. Since one is shaded below and the other above, the assertion that both are shaded below is incorrect. Therefore, this statement is false. Understanding the direction of shading is paramount to correctly identifying the solution set for an inequality and, by extension, the precise overlapping region for a system.

By meticulously breaking down each component of these linear inequalities and their graphical representations, we've not only answered the specific questions but also gained a deeper, more comprehensive understanding of how systems of inequalities work. It's a powerful tool for modeling real-world constraints and finding valid ranges of solutions. Keep practicing, and you'll be a graphing inequalities pro in no time!

To enrich your understanding of linear inequalities and their applications, consider exploring additional resources from reputable mathematics education sites. For more detailed explanations on graphing linear inequalities and systems of inequalities, check out Khan Academy's Algebra content or Purplemath's step-by-step guides. These resources offer fantastic examples and practice problems to solidify your knowledge.