Cara Mudah Menggambar Daerah Penyelesaian Sistem Pertidaksamaan

by Dimemap Team 64 views

Hey guys! Are you ready to dive into the world of inequalities and coordinate planes? In this article, we'll break down how to graph the solution regions for a system of linear inequalities. This is super important stuff in algebra, and trust me, it's not as scary as it looks. We'll be working with a set of inequalities and finding the area where all of them are true at the same time. Think of it like a treasure hunt where the treasure is the solution! We'll be using the following inequalities:

  • (I) 4x + 3y > -24
  • (II) x - 2y ≤ 10
  • (III) x > -4
  • (IV) y ≤ 1

Let's get started, shall we? This tutorial will guide you step by step in visualizing your solutions. Make sure to follow the instructions thoroughly, so you can fully understand the material. If you can understand this material, you will be able to solve many problems.

Step-by-Step Guide to Graphing Linear Inequalities

First things first, before we start solving, let's understand the basics of graphing linear inequalities. Remember, a linear inequality is just like a linear equation (like y = mx + b), but instead of an equals sign, we have an inequality sign such as > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). The key to graphing these bad boys is to treat them like equations initially. We'll start with the first inequality.

Inequality (I): 4x + 3y > -24

Our first step is to treat this inequality as an equation: 4x + 3y = -24. This is a linear equation, and we can graph it by finding two points. The easiest points to find are the x-intercept (where y = 0) and the y-intercept (where x = 0).

  • Finding the x-intercept: Set y = 0. Then, 4x + 3(0) = -24, which simplifies to 4x = -24. Dividing both sides by 4, we get x = -6. So, our x-intercept is (-6, 0).
  • Finding the y-intercept: Set x = 0. Then, 4(0) + 3y = -24, which simplifies to 3y = -24. Dividing both sides by 3, we get y = -8. So, our y-intercept is (0, -8).

Now, plot these two points on your coordinate plane and draw a dashed line through them. Why dashed? Because the inequality is “greater than” (>), not “greater than or equal to” (≥). A dashed line means that the points on the line are not included in the solution.

Next, we need to figure out which side of the line represents the solution. Choose a test point that's not on the line. The easiest one is usually (0, 0). Substitute x = 0 and y = 0 into the original inequality: 4(0) + 3(0) > -24. This simplifies to 0 > -24, which is true. Since the test point (0, 0) satisfies the inequality, the solution region is the side of the line that includes (0, 0). Shade this area. This means all the points in that region will satisfy our inequality. This step helps us determine the areas of the graph.

Inequality (II): x - 2y ≤ 10

Let’s move on to the next one! Treat x - 2y ≤ 10 as an equation: x - 2y = 10.

  • Finding the x-intercept: Set y = 0. Then, x - 2(0) = 10, which simplifies to x = 10. So, our x-intercept is (10, 0).
  • Finding the y-intercept: Set x = 0. Then, 0 - 2y = 10, which simplifies to -2y = 10. Dividing both sides by -2, we get y = -5. So, our y-intercept is (0, -5).

Plot these points and draw a solid line through them. Why solid this time? Because the inequality is “less than or equal to” (≤), which means the points on the line are included in the solution. This is very important, because if the line is not solid, the answer may be wrong.

Again, we need to find the solution region. Use the test point (0, 0): 0 - 2(0) ≤ 10. This simplifies to 0 ≤ 10, which is true. Shade the side of the line that includes (0, 0). Keep in mind that the solution to a system of inequalities is where all shaded regions overlap. Make sure you don't make a mistake when shading the areas.

Inequality (III): x > -4

This one is different! This inequality only involves x. This is the easiest one! This is a vertical line. Think of it this way: for every point on the line, the x-coordinate is always -4. So, we draw a vertical dashed line at x = -4 (dashed because it's “greater than” not “greater than or equal to”).

The solution is all the values of x greater than -4, so we shade the region to the right of the line.

Inequality (IV): y ≤ 1

This one is another special case, but this time, it’s a horizontal line. The line is located at y = 1, and since it is “less than or equal to”, the line is solid. Shade the area below the line, because it's where y values are less than or equal to 1. Similar to the previous inequality, it is very important to not make any mistakes.

Finding the Solution Region

Alright, you've graphed all the inequalities! Now, the solution to the system is the region where all the shaded areas overlap. It's like finding a sweet spot where all the conditions are met. This is where all the shaded regions intersect each other. This is the place where everything is true. This area represents all the solutions that satisfy all of the given inequalities. This is why we are doing all these steps. This is the answer to your problems.

  • Visualize the Overlap: Imagine that each inequality is a different color. The solution is where all the colors mix together.
  • Identify the Boundaries: Look at the lines and see which ones are solid and which ones are dashed. These boundaries define the solution region.
  • Test Points (Optional): You can pick a point within the solution region and test it in all the original inequalities to make sure it works.

Tips for Success

  • Be Organized: Keep your work neat and clearly labeled. It's easy to get lost if things are messy.
  • Use Different Colors: This helps you visually track each inequality’s solution region.
  • Double-Check Your Work: Make sure you've correctly found the intercepts, drawn the lines correctly (solid or dashed), and shaded the correct regions.
  • Practice, Practice, Practice: The more you practice, the easier it will become. Try different sets of inequalities.
  • Online Tools: Use online graphing calculators (like Desmos) to check your work and visualize the graphs. It is really easy to use.

Conclusion

And there you have it! You've learned how to graph the solution region of a system of linear inequalities. It might seem like a lot at first, but with practice, it becomes second nature. Remember to treat each inequality as an equation, find the intercepts, draw the lines (solid or dashed), shade the solution region, and then find the overlap. You are now ready to solve more complex problems! This knowledge is incredibly useful in various areas of mathematics, from optimization problems to real-world applications. Good luck, and happy graphing, guys! Feel free to ask questions if you are confused, because it can be confusing at first.