Graphing Inequalities: Finding The Solution Set Of 6x + 4y < 12

Hey everyone! Today, we're diving into the world of graphing inequalities, and we're going to figure out which graph correctly represents the solution set for the inequality 6x + 4y < 12. Don't worry, it sounds a bit complicated, but I promise we'll break it down step by step, and by the end, you'll be a pro at this. So, grab your pencils and let's get started!

Understanding the Basics: Inequality vs. Equation

First things first, let's make sure we're on the same page about the difference between an equation and an inequality. An equation uses an equals sign (=), showing that two expressions are exactly the same. For example, 2x + 3 = 7 is an equation. Our main focus is inequalities, which use signs like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). The inequality 6x + 4y < 12 means that the expression on the left side is less than the number on the right side. The solution set to an inequality isn't just one point like it is for an equation, but rather a whole region on the coordinate plane. Think of it like this: an equation is a single road, while an inequality is an entire neighborhood.

Now, let's think about how to tackle 6x + 4y < 12. The inequality describes all the points (x, y) that, when plugged into the expression 6x + 4y, result in a value less than 12. To visualize this, we graph the related equation: 6x + 4y = 12. This equation is a line, and we use it as a boundary to divide the coordinate plane into two regions. One region represents the solution set for the inequality, and the other does not.

To graph the equation, one way is to find the x and y intercepts. Let's start with the x-intercept. Set y = 0 in the equation 6x + 4y = 12. We get 6x + 4(0) = 12, which simplifies to 6x = 12. Dividing both sides by 6, we find x = 2. So, the x-intercept is the point (2, 0). Next, let's find the y-intercept. This time, we set x = 0 in the equation 6x + 4y = 12. This gives us 6(0) + 4y = 12, which simplifies to 4y = 12. Dividing both sides by 4, we get y = 3. The y-intercept is the point (0, 3). So, to graph the line, we can just plot these two points on the coordinate plane and draw a straight line through them. This line acts as the boundary for our inequality. The intercepts provide crucial points for graphing, ensuring accuracy and enabling us to identify the correct solution set visually.

From Equation to Inequality: Shading the Correct Region

Okay, we have our line, but remember, the inequality is 6x + 4y < 12, not 6x + 4y = 12. This means we need to figure out which side of the line represents the solution set. To do this, we can pick a test point—any point that isn't on the line. The easiest one to use is usually (0, 0). Now, plug the x and y values from our test point into the original inequality 6x + 4y < 12. So, we get 6(0) + 4(0) < 12, which simplifies to 0 < 12. Is this true? Yes! Since 0 is indeed less than 12, this means the test point (0, 0) is part of the solution set. Therefore, we shade the side of the line that includes the point (0, 0).

If the inequality had been 6x + 4y > 12, our test point (0, 0) would have given us 0 > 12, which is false. In that case, we would have shaded the other side of the line, the side that does not include (0, 0). The inequality sign determines how we represent the boundary line. If the inequality is < or > (strictly less than or greater than), the boundary line is dashed to indicate that the points on the line itself are not included in the solution set. However, if the inequality is ≤ or ≥ (less than or equal to, or greater than or equal to), the boundary line is solid, meaning the points on the line are included in the solution set. This distinction is crucial for accurately representing the solution set.

Remember, when graphing an inequality, it's not just about drawing a line; it's about identifying a region. That region includes all the points that satisfy the inequality. The shading visually indicates which side of the line contains the solution set. The shaded region represents all the ordered pairs (x, y) that, when substituted into the inequality, make it true. This is the graphical representation of the solution to the inequality.

Step-by-Step Guide to Graphing the Inequality

Let's summarize the process. First, let's convert the inequality to the corresponding equation form. For the inequality 6x + 4y < 12, the corresponding equation is 6x + 4y = 12. Next, find the x and y intercepts of the equation. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. Then, plot the intercepts on the coordinate plane and draw the line. Note that, since our inequality is strictly less than, we draw the line as dashed. Alternatively, you could rewrite the equation in slope-intercept form y = mx + b, which is y = (-3/2)x + 3. From this equation, we know the y-intercept is 3, and the slope is -3/2. Starting from the y-intercept, use the slope to find other points and then draw the line.

Now, select a test point. The easiest test point is usually (0, 0), as long as it's not on the line. Substitute the x and y values from your test point into the inequality 6x + 4y < 12. If the resulting statement is true, shade the region that contains your test point. If the statement is false, shade the other region. For 6x + 4y < 12, our test point (0, 0) makes the inequality true, so we shade the region containing (0, 0). This shaded region represents the solution set for the inequality. Any point within this shaded region, when plugged back into the original inequality 6x + 4y < 12, will result in a true statement. And there you have it, you've successfully graphed the inequality and found its solution set!

Real-World Applications: Where Inequalities Come in Handy

Okay, so why is this important, right? Well, inequalities show up everywhere! They're super useful for modeling real-world situations. For instance, imagine you're planning a bake sale. You have a limited budget and want to bake cookies and brownies. Let x represent the number of cookies, which cost $2 each, and let y represent the number of brownies, which cost $3 each. If you have $12 to spend, then the inequality might be 2x + 3y ≤ 12. This inequality helps you figure out the different combinations of cookies and brownies you can bake without exceeding your budget. The solution set would show all possible combinations. We can solve by graphing to see this solution set.

Another example could be a situation involving resource allocation. Imagine a company has a limited amount of raw materials to produce two different products. The inequality could represent the constraints, such as the maximum amount of each product that can be produced given the available resources. Inequalities are also crucial in fields such as linear programming, optimization problems, and economics, where they help to establish constraints and find optimal solutions. Inequalities are tools that allow us to model and solve a wide range of problems, from personal finance to business operations. Understanding them allows us to make informed decisions based on realistic constraints.

Common Mistakes to Avoid

Let's cover a few common pitfalls to make sure you're on the right track. One mistake is forgetting to use a dashed or solid line correctly. Remember: a dashed line for < or >, and a solid line for ≤ or ≥. Another mistake is shading the wrong region. Always use a test point to ensure you're shading the correct side of the line. Make sure you correctly solve for the intercepts. Carefully compute your solution to make sure that the points where the line crosses the axis are correct. Double-check your calculations when plugging in the test point to make sure your solution is valid. A simple math error can lead to a wrong answer. These are common errors, but by being careful and double-checking your work, you can easily avoid them. Make it a habit to label your axes and intercepts clearly. Clear labeling makes it easier to understand and interpret your graphs, and also helps to communicate your solution effectively.

Conclusion: You Got This!

Alright, guys, you've made it! We've covered the ins and outs of graphing inequalities like 6x + 4y < 12. You know how to identify the boundary line, choose a test point, and shade the correct region. Remember, practice makes perfect. Try graphing other inequalities. The more you practice, the more comfortable you'll become. Keep at it, and you'll be acing those math problems in no time. You have the knowledge and tools to succeed. Keep up the great work, and happy graphing! Remember to practice, and don't be afraid to ask for help if you get stuck. You've got this!