Graphing Linear Inequalities 2x - 3y < 12 Solved Step By Step

by Luna Greco 62 views

Hey guys! Let's dive into the world of linear inequalities and learn how to graph them. It might sound intimidating, but trust me, it's totally manageable. We're going to break down the inequality 2x - 3y < 12 and figure out how to represent it visually on a graph. So, buckle up, and let's get started!

Understanding Linear Inequalities

Before we jump into graphing, let's make sure we're all on the same page about what a linear inequality actually is. Think of it as a cousin of the good old linear equation (like y = mx + b), but instead of an equals sign, we have inequality symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). These symbols tell us that we're dealing with a range of possible solutions, not just a single line.

Linear inequalities are all about defining regions on a graph. The solutions to a linear inequality aren't just points on a line; they're all the points within a certain area. This area is bounded by a line, which we call the boundary line. The inequality symbol tells us whether the boundary line is included in the solution (solid line for ≤ and ≥) or excluded (dashed line for < and >). Understanding this difference is super important, because it helps us accurately represent the solution set on our graph. Think of it this way: the solid line is like saying "Hey, these points on the line are also part of the club!", while the dashed line is like saying "This line is the VIP rope, but the points on it aren't invited to the party."

Our main goal here is to graph the linear inequality 2x - 3y < 12. This means we need to find all the points (x, y) that make this statement true. We'll do this by first graphing the boundary line and then shading the region that contains the solutions. It's like drawing a map where the shaded area marks all the hidden treasures (the solutions!). The boundary line acts as our guide, showing us where the treasure hunt begins and ends. So, let’s get our shovels ready and start digging into the steps involved!

Step-by-Step Guide to Graphing 2x - 3y < 12

Alright, let's break down the process of graphing 2x - 3y < 12 into easy-to-follow steps. Don't worry, we'll take it slow and make sure everything clicks.

Step 1: Convert to Slope-Intercept Form (Optional but Helpful)

This step is optional, but I highly recommend it because it makes graphing the line much easier. Slope-intercept form is that friendly y = mx + b format we all know and love, where m is the slope and b is the y-intercept. It’s like having a clear roadmap to plot our line.

Let’s rearrange 2x - 3y < 12 to slope-intercept form. First, we want to isolate the y term. We can do this by subtracting 2x from both sides of the inequality:

-3y < -2x + 12

Now, we need to get y by itself. Divide both sides by -3. Remember! When we multiply or divide an inequality by a negative number, we have to flip the inequality sign. This is a crucial rule, so don't forget it! It's like a secret handshake in the world of inequalities.

So, we get:

y > (2/3)x - 4

Now our inequality is in slope-intercept form! We can clearly see that the slope (m) is 2/3 and the y-intercept (b) is -4. Having this information is like having the GPS coordinates for our line – we know exactly where to start and how to move.

Step 2: Graph the Boundary Line

Now that we have our inequality in slope-intercept form, it's time to draw the boundary line. The boundary line is the line that separates the solutions from the non-solutions. It’s the edge of our solution zone.

Using our slope-intercept form y > (2/3)x - 4, we know the slope is 2/3 and the y-intercept is -4. Let's plot the y-intercept first. This is the point where the line crosses the y-axis, so we'll put a point at (0, -4). It’s our starting point, our home base.

Next, we'll use the slope to find another point on the line. The slope 2/3 means “rise 2, run 3.” So, from our y-intercept, we'll go up 2 units and then right 3 units. This will give us a second point on the line. We can keep using the slope to find more points if we want, but two points are enough to draw a line.

Now, here's the important part: Because our inequality is > (greater than) and not ≥ (greater than or equal to), we'll draw a dashed line. A dashed line indicates that the points on the line are not included in the solution. It's like an invisible barrier – we know it's there, but we can't cross it. If our inequality had been ≥, we would have drawn a solid line, meaning the points on the line are part of the solution. It’s all about being precise and paying attention to those little details!

Step 3: Shade the Correct Region

We've got our boundary line drawn, but we're not done yet! Now we need to figure out which side of the line represents the solutions to our inequality. This is where the shading comes in. We're going to shade the region that contains all the points that make y > (2/3)x - 4 true. It's like coloring in the area where the treasure is hidden.

To figure out which region to shade, we can use a test point. A test point is any point that is not on the boundary line. The easiest test point to use is usually the origin, (0, 0), unless the line goes through the origin. Let's plug (0, 0) into our inequality and see what happens:

0 > (2/3)(0) - 4

0 > -4

Is this statement true? Yes, it is! 0 is indeed greater than -4. This means that the point (0, 0) is a solution to our inequality. Therefore, we need to shade the region that contains (0, 0). It’s like the test point gave us the secret password to unlock the solution zone.

If the test point had made the inequality false, we would have shaded the other region. So, the side we shade depends entirely on whether the test point satisfies the inequality. Grab your colored pencils (or your digital shading tool) and fill in the correct area. You’re one step closer to graphing that inequality like a pro!

Putting It All Together: The Graph of 2x - 3y < 12

Okay, let's recap everything we've done and visualize the final graph of 2x - 3y < 12. We've gone through the steps, and now it's time to see the big picture.

  1. We started with the inequality 2x - 3y < 12.
  2. We converted it to slope-intercept form: y > (2/3)x - 4.
  3. We graphed the boundary line using the slope (2/3) and y-intercept (-4). Remember, it's a dashed line because our inequality is >.
  4. We used the test point (0, 0) and found that it satisfied the inequality.
  5. We shaded the region above the dashed line, because that's where all the solutions lie.

So, our final graph consists of a dashed line with a slope of 2/3 and a y-intercept of -4, and the region above the line is shaded. This shaded area represents all the points (x, y) that make the inequality 2x - 3y < 12 true. It's like we've created a visual map of the solutions!

When you look at the graph, imagine any point in the shaded region. If you were to plug those x and y values into the inequality, it would hold true. On the other hand, any point outside the shaded region (or on the dashed line) would not satisfy the inequality. This is the beauty of graphing inequalities – it gives us a clear and intuitive way to understand the solution set.

Common Mistakes to Avoid

Graphing linear inequalities is pretty straightforward once you get the hang of it, but there are a few common mistakes people often make. Let's talk about them so you can steer clear and ace those graphs!

Forgetting to Flip the Inequality Sign

This is a big one! Remember, when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. If you forget this step, you'll end up shading the wrong region, and your graph will be incorrect. It’s like forgetting to add the secret ingredient to a recipe – the final result just won’t be right.

Using the Wrong Type of Line

It's crucial to use a dashed line for inequalities with < or > and a solid line for inequalities with ≤ or ≥. The type of line tells us whether the points on the line are included in the solution or not. Think of it as using the right kind of tool for the job – a wrench won’t work if you need a screwdriver.

Choosing a Test Point on the Line

Your test point must not be on the boundary line. If you choose a point on the line, it won't tell you which side to shade. The whole point of the test point is to help you determine which region contains the solutions, so make sure it's clearly on one side or the other. It’s like trying to decide between two paths by standing right on the dividing line – you need to step to one side to see where it leads.

Shading the Wrong Region

This usually happens when you either forget to flip the inequality sign or misinterpret the test point. Double-check your work to make sure you're shading the region that contains the solutions. It’s like reading a map upside down – you might end up going in the opposite direction!

Not Converting to Slope-Intercept Form

While not strictly a mistake, not converting to slope-intercept form can make graphing the line much harder. Slope-intercept form gives you the slope and y-intercept, which are super helpful for plotting the line accurately. It’s like trying to build a house without a blueprint – you can probably do it, but it’ll be much easier with a plan.

By keeping these common mistakes in mind, you'll be well on your way to graphing linear inequalities like a pro. Remember, practice makes perfect, so don't be afraid to try out lots of examples!

Practice Makes Perfect: Examples and Exercises

Now that we've covered the steps and the common pitfalls, it's time to put your knowledge to the test! Practice is the key to mastering any skill, and graphing linear inequalities is no exception. Let's work through a few examples and then give you some exercises to try on your own.

Example 1: Graphing y ≤ -x + 3

  1. Slope-intercept form: The inequality is already in slope-intercept form! We can see that the slope is -1 and the y-intercept is 3.
  2. Boundary line: We'll draw a solid line because the inequality is ≤. Plot the y-intercept at (0, 3). Use the slope -1 (which is the same as -1/1) to find another point: go down 1 unit and right 1 unit. Connect the points with a solid line.
  3. Test point: Let's use (0, 0). Plug it into the inequality: 0 ≤ -0 + 3 0 ≤ 3 This is true! So, we shade the region that contains (0, 0), which is the region below the line.

Example 2: Graphing 3x + 2y > 6

  1. Slope-intercept form: Let's convert the inequality: 2y > -3x + 6 y > (-3/2)x + 3 Our slope is -3/2 and the y-intercept is 3.
  2. Boundary line: We'll draw a dashed line because the inequality is >. Plot the y-intercept at (0, 3). Use the slope -3/2 to find another point: go down 3 units and right 2 units. Connect the points with a dashed line.
  3. Test point: Let's use (0, 0) again: 0 > (-3/2)(0) + 3 0 > 3 This is false! So, we shade the region that doesn't contain (0, 0), which is the region above the line.

Exercises for You to Try

Now it's your turn! Graph the following inequalities. Remember to follow the steps we've discussed, and don't be afraid to make mistakes – that's how we learn!

  1. y < 2x - 1
  2. x + y ≥ 4
  3. 2x - y ≤ 5
  4. y > -3
  5. x ≤ 1

Check your answers by graphing them on a graphing calculator or an online graphing tool. The more you practice, the more confident you'll become in your graphing abilities. Happy graphing!

Conclusion: You've Got This!

Wow, we've covered a lot! We've explored the world of linear inequalities, learned how to graph them step-by-step, discussed common mistakes to avoid, and even worked through some examples. You've come a long way, and you should be proud of your progress!

Graphing linear inequalities might seem tricky at first, but the key is to break it down into manageable steps. Remember:

  • Convert to slope-intercept form (y = mx + b) if needed.
  • Graph the boundary line (dashed or solid).
  • Choose a test point and shade the correct region.

By following these steps and practicing regularly, you'll be able to tackle any linear inequality that comes your way. Think of each graph as a puzzle – a fun challenge that you can solve with the tools and knowledge you've gained. You've got this!

So, the next time you encounter a linear inequality, don't shy away. Embrace the challenge, grab your graph paper (or your favorite graphing app), and start plotting those lines and shading those regions. You'll be amazed at how quickly you improve. Keep practicing, stay curious, and remember that every graph you create is a step closer to mastering the art of linear inequalities. You're doing great, guys! Keep up the awesome work!