Graphing The Inequality 10x - 5y ≤ 50 A Step-by-Step Guide

Jul 16, 2025 by ADMIN 59 views

Graphing Linear Inequalities: A Comprehensive Guide to Understanding 10x - 5y ≤ 50

Hey guys! Today, we're diving deep into the world of graphing linear inequalities, specifically focusing on the inequality 10x - 5y ≤ 50. Graphing inequalities might seem tricky at first, but trust me, once you understand the core concepts, it becomes super straightforward. We'll break down each step, making sure you not only grasp how to graph this particular inequality but also understand the underlying principles so you can tackle any similar problem with confidence. So, let's jump right in and unlock the secrets of graphing inequalities!

Understanding Linear Inequalities

Before we even think about plotting lines and shading regions, let's make sure we're all on the same page about what a linear inequality actually is. At its heart, a linear inequality is simply a mathematical statement that compares two expressions using inequality symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike linear equations, which have a single solution (or a set of solutions), linear inequalities have a range of solutions. This range is represented graphically as a shaded region on the coordinate plane. The inequality 10x - 5y ≤ 50 is a classic example. It states that the expression 10x - 5y must be less than or equal to 50. This means that any point (x, y) that satisfies this condition is a solution to the inequality. Our goal is to visually represent all these solutions on a graph.

Linear inequalities are super useful in real-world applications. Imagine you're budgeting for a party. You have a certain amount of money to spend on food and drinks, and each item has a different cost. A linear inequality can help you determine the different combinations of food and drinks you can buy without exceeding your budget. Or, think about constraints in manufacturing. A company might have limitations on the amount of resources or time available to produce goods. Linear inequalities can model these constraints and help optimize production. Understanding how to work with these inequalities, including graphing them, is a valuable skill in various fields.

Now, let's dive deeper into the specific components of our inequality, 10x - 5y ≤ 50. Notice that it involves two variables, x and y, and the highest power of these variables is 1. This is what makes it a linear inequality. The coefficients (the numbers in front of the variables) and the constant term (the number on the right side of the inequality) determine the position and orientation of the boundary line, which we'll talk about next. The inequality symbol (≤ in this case) tells us which side of the line represents the solution region. So, each part of the inequality plays a crucial role in determining its graph, and understanding these roles is key to graphing inequalities accurately. Let's move on to the next step: simplifying the inequality.

Step 1: Simplifying the Inequality

Okay, first things first, let's simplify our inequality 10x - 5y ≤ 50 to make it easier to work with. Simplifying inequalities is a smart move because it reduces the chances of making mistakes later on when we're plotting points and drawing lines. The goal here is to get the inequality into a more manageable form, ideally something that resembles the slope-intercept form of a linear equation (y = mx + b), which you might remember from algebra class. This form makes it super easy to identify the slope and y-intercept, which are essential for graphing the line.

So, how do we simplify? We use the same algebraic principles we use for equations, with one important exception that we'll discuss in a bit. We can add, subtract, multiply, or divide both sides of the inequality by a constant, just like we do with equations. Our aim is to isolate the 'y' term on one side. In this case, we can start by dividing both sides of the inequality by 5. This gives us:

(10x - 5y) / 5 ≤ 50 / 5

Which simplifies to:

2x - y ≤ 10

Now, let's isolate 'y'. We can subtract 2x from both sides:

-y ≤ -2x + 10

Here comes the crucial part: when we multiply or divide both sides of an inequality by a negative number, we need to flip the inequality sign. This is a super important rule to remember! It's because multiplying or dividing by a negative number reverses the order of the numbers on the number line. So, to get 'y' by itself, we need to multiply both sides by -1. This gives us:

(-1) * (-y) ≥ (-1) * (-2x + 10)

Which simplifies to:

y ≥ 2x - 10

Ta-da! We've successfully simplified the inequality. This form, y ≥ 2x - 10, is much easier to interpret. We can now clearly see that the slope of the boundary line is 2 and the y-intercept is -10. The '≥' symbol tells us that we're interested in the region where y is greater than or equal to 2x - 10. Next up, we'll learn how to graph the boundary line itself.

Step 2: Graphing the Boundary Line

Alright, now that we've simplified our inequality to y ≥ 2x - 10, it's time to graph the boundary line. The boundary line is the line that separates the region of solutions from the region of non-solutions. It's like the fence that marks the edge of our solution set. In our case, the boundary line is represented by the equation y = 2x - 10. Notice that we've simply replaced the inequality symbol (≥) with an equals sign (=).

To graph this line, we can use a few different methods. One common approach is to use the slope-intercept form, which we already have! Remember, the slope-intercept form is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. From our simplified inequality, we know that the slope (m) is 2 and the y-intercept (b) is -10. This means the line crosses the y-axis at the point (0, -10), and for every 1 unit we move to the right, the line goes up 2 units.

We can plot the y-intercept (0, -10) on our graph. Then, using the slope of 2, we can find another point on the line. If we move 1 unit to the right from (0, -10), we go up 2 units, landing us at the point (1, -8). We can repeat this process to find a few more points, like (2, -6) and (3, -4). Once we have a few points, we can draw a straight line through them. This line is our boundary line.

But wait, there's one more crucial detail! We need to decide whether to draw a solid line or a dashed line. This depends on the inequality symbol in our original inequality. If the inequality symbol is ≤ or ≥ (like in our case), we draw a solid line. This indicates that the points on the line are also solutions to the inequality. If the inequality symbol is < or >, we draw a dashed line. This means the points on the line are not included in the solution set. Since our inequality is y ≥ 2x - 10, we'll draw a solid line. This is because the 'equal to' part of the '≥' means that points on the line satisfy the inequality.

So, we've plotted the y-intercept, used the slope to find other points, and drawn a solid line through them. Our boundary line is now graphed! We're one step closer to visualizing the solution set. Next, we need to figure out which side of the line to shade. This is where the test point method comes in handy.

Step 3: Shading the Correct Region

Okay, we've got our boundary line graphed, which is half the battle. Now, the crucial question is: which side of the line do we shade? The shaded region represents all the points (x, y) that satisfy our inequality, y ≥ 2x - 10. To figure out which side to shade, we use a nifty trick called the test point method. This method involves picking a point that is not on the line and plugging its coordinates into the original inequality. The result will tell us whether that point is a solution or not, and thus, which side of the line contains the solutions.

The easiest point to use as a test point is usually the origin, (0, 0), as long as the boundary line doesn't pass through it. In our case, the line y = 2x - 10 doesn't go through the origin, so (0, 0) is a perfect choice. Let's plug x = 0 and y = 0 into our inequality:

0 ≥ 2(0) - 10

This simplifies to:

0 ≥ -10

Is this statement true? Yes, 0 is indeed greater than or equal to -10. This means that the point (0, 0) is a solution to the inequality. Therefore, we need to shade the side of the line that contains the point (0, 0).

If, on the other hand, we had plugged in (0, 0) and gotten a false statement, we would shade the other side of the line. The test point method is a reliable way to determine the correct shading region every time. It's like a compass that guides us to the solution set.

So, we've plugged in our test point, determined that it satisfies the inequality, and now we know which side of the line to shade. Grab your pencil and lightly shade the region that contains the point (0, 0). This shaded region represents all the points (x, y) that make the inequality y ≥ 2x - 10 true. In other words, every point within the shaded area is a solution to our original inequality, 10x - 5y ≤ 50.

We've successfully graphed the inequality! Let's take a moment to recap the steps we took and solidify our understanding.

Recapping the Steps

Alright, let's quickly recap the steps we took to graph the inequality 10x - 5y ≤ 50. This will help solidify the process in your mind and make you a graphing whiz in no time!

Simplify the Inequality: Our first step was to simplify the inequality to make it easier to work with. We divided both sides by 5, subtracted 2x from both sides, and then multiplied both sides by -1 (remembering to flip the inequality sign!). This gave us the simplified form y ≥ 2x - 10. Simplifying is key because it makes the next steps much smoother.
Graph the Boundary Line: Next, we graphed the boundary line, which is the line that separates the solution region from the non-solution region. To do this, we treated the inequality as an equation (y = 2x - 10) and used the slope-intercept form to identify the slope (2) and y-intercept (-10). We plotted the y-intercept and used the slope to find other points on the line. We drew a solid line because our inequality symbol was ≥ (including the 'equal to' part).
Shade the Correct Region: Finally, we needed to determine which side of the line to shade. We used the test point method, choosing the origin (0, 0) as our test point. We plugged the coordinates of the test point into the original inequality and found that it satisfied the inequality. This meant we needed to shade the side of the line that contained the origin. And just like that, we had our solution set!

These three steps – simplify, graph the boundary line, and shade the correct region – are the foundation for graphing any linear inequality. The more you practice, the more comfortable you'll become with each step. Now that we've recapped the process, let's tackle some common mistakes people make when graphing inequalities, so you can avoid them.

Common Mistakes to Avoid

Graphing inequalities can be a bit tricky, and it's easy to make small mistakes that can lead to a completely wrong graph. But don't worry, we're here to help you steer clear of those pitfalls! Let's talk about some common mistakes people make and how to avoid them.

Forgetting to Flip the Inequality Sign: This is probably the most common mistake. Remember, whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. If you forget to do this, you'll end up shading the wrong region, leading to an incorrect solution set. Double-check this step every time you simplify an inequality!
Drawing the Wrong Type of Line: It's crucial to pay attention to the inequality symbol when deciding whether to draw a solid or dashed line. A solid line indicates that the points on the line are included in the solution set (≤ or ≥), while a dashed line indicates that they are not (< or >). Drawing the wrong type of line will misrepresent the solutions to the inequality.
Choosing a Test Point on the Line: The test point method is a powerful tool, but it only works if you choose a point that is not on the boundary line. If you accidentally pick a point on the line, you won't get a clear indication of which side to shade. Always choose a point that is clearly on one side or the other.
Shading the Wrong Region: Even if you do everything else correctly, you can still end up with the wrong graph if you shade the wrong region. This usually happens if you misinterpret the result of the test point. Remember, if the test point satisfies the inequality, shade the side containing the test point. If it doesn't, shade the other side.
Not Simplifying the Inequality: Trying to graph an inequality without simplifying it first can be a recipe for disaster. Complex inequalities are harder to work with, and the chances of making a mistake increase. Always simplify the inequality as much as possible before you start graphing.

By being aware of these common mistakes, you can avoid them and graph inequalities accurately. Practice makes perfect, so keep working on these skills, and you'll become a pro in no time!

Conclusion

Woohoo! You've made it to the end of our comprehensive guide on graphing the inequality 10x - 5y ≤ 50. We've covered a lot of ground, from understanding the basics of linear inequalities to simplifying, graphing the boundary line, and shading the correct region. You've learned how to use the test point method and how to avoid common mistakes. Give yourself a pat on the back – you've earned it!

Graphing inequalities is a fundamental skill in mathematics with applications in various fields. By mastering this skill, you're not only expanding your mathematical knowledge but also equipping yourself with a valuable tool for problem-solving in the real world. So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is full of fascinating concepts waiting to be discovered, and you're well on your way to becoming a math whiz!