Solving System Of Inequalities: A Step-by-Step Guide
Hey guys! Today, we're diving into the fascinating world of system of inequalities! This topic might seem a bit intimidating at first, but trust me, it’s super manageable once you understand the basic steps. We're going to break down a specific problem and walk through it together, making sure you grasp each concept along the way. So, let's jump right into it and make math a little less scary, shall we?
Understanding System of Inequalities
Before we tackle the main problem, let's quickly recap what system of inequalities actually means. Think of it as a set of two or more inequalities that you need to solve simultaneously. Unlike equations that have precise solutions, inequalities deal with ranges of values. When we have a system, we're looking for the region on a graph where all the inequalities are true at the same time. This is often represented by a shaded area on the coordinate plane. So, the solution to a system of inequalities isn't just one point, but rather a whole bunch of points that fit the criteria for all inequalities in the system.
Linear Inequalities
The core of our discussion revolves around linear inequalities, which are inequalities involving linear expressions. A linear expression is something that, when graphed, forms a straight line. Examples include expressions like x + 2y or -x + 4y. The inequality part comes from symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). These symbols tell us that the values can be less than, greater than, or equal to a certain amount, rather than just a specific number. Remember, the equality aspect in ≤ and ≥ means the line itself is included in the solution, whereas < and > mean the line is a boundary but not part of the solution. We'll see how this plays out graphically in a bit!
Problem Statement: Our System of Inequalities
Okay, now let's get to the nitty-gritty! We're going to tackle this specific system of inequalities:
Our mission is to find all the (x, y) pairs that satisfy both of these inequalities. In simpler terms, we need to figure out the region on the coordinate plane where both inequalities hold true. This involves a few key steps, including graphing each inequality and identifying the overlapping region. Think of it as a treasure hunt, where the solution is the hidden treasure that satisfies all the clues (inequalities). Let's start deciphering those clues, shall we?
Step 1: Graphing the First Inequality ()
First up, we'll graph the inequality x + 2y ≤ 6. To do this, we'll follow a few sub-steps to make sure we get it right. Graphing inequalities isn't too tricky, but it requires attention to detail, especially when it comes to deciding whether to use a solid or dashed line and which side to shade. So, let’s take it slow and steady!
Convert the Inequality to an Equation
Our first move is to treat the inequality as an equation. So, we change x + 2y ≤ 6 to x + 2y = 6. Why do we do this? Because the equation represents the boundary line of our inequality. This line is the divider between the region where the inequality is true and the region where it's false. Finding this line is crucial for visualizing the solution set. It’s like setting up the borders of our treasure map!
Find Two Points on the Line
To draw the line, we need at least two points. The easiest way to find these points is often by setting x to 0 and solving for y, and then setting y to 0 and solving for x. These give us the y-intercept and x-intercept, respectively. Let's do it:
- Set x = 0: 0 + 2y = 6 2y = 6 y = 3 So, one point is (0, 3).
- Set y = 0: x + 2(0) = 6 x = 6 So, another point is (6, 0).
Now we have two points, (0, 3) and (6, 0), which we can use to draw our line.
Draw the Line (Solid or Dashed?)
Here’s a crucial step: deciding whether to draw a solid or dashed line. This depends on the inequality symbol. If we have ≤ or ≥, we use a solid line because the points on the line are included in the solution. If we have < or >, we use a dashed line because the points on the line are not part of the solution. They're just the boundary. In our case, we have ≤, so we'll draw a solid line through (0, 3) and (6, 0). This solid line tells us that the points on the line itself are part of the solution.
Shade the Correct Region
Now, for the final touch: shading the correct side of the line. The line divides the coordinate plane into two regions, and we need to figure out which region satisfies the inequality x + 2y ≤ 6. To do this, we can use a test point. The easiest test point is often (0, 0), as long as the line doesn't pass through it. Let's plug (0, 0) into our inequality:
0 + 2(0) ≤ 6 0 ≤ 6
This is true! Since (0, 0) makes the inequality true, we shade the region that contains (0, 0). If it were false, we'd shade the other side. So, we've now graphed our first inequality: a solid line through (0, 3) and (6, 0), with the region below the line shaded.
Step 2: Graphing the Second Inequality (-x + 4y > 12)
Alright, let's move on to the second inequality: -x + 4y > 12. We'll follow the same steps as before, making sure we pay close attention to whether the line should be solid or dashed and which side we need to shade.
Convert the Inequality to an Equation
Just like before, we start by changing the inequality to an equation: -x + 4y = 12. This gives us the boundary line for our second inequality. It's like marking another clue on our treasure map!
Find Two Points on the Line
Let's find two points on this line. Again, setting x and y to 0 is a good starting point:
- Set x = 0: -0 + 4y = 12 4y = 12 y = 3 So, one point is (0, 3).
- Set y = 0: -x + 4(0) = 12 -x = 12 x = -12 So, another point is (-12, 0).
We've got our two points: (0, 3) and (-12, 0).
Draw the Line (Solid or Dashed?)
Now, the crucial question: solid or dashed line? Our inequality is -x + 4y > 12, which uses the > symbol. This means the points on the line are not included in the solution. So, we'll draw a dashed line through (0, 3) and (-12, 0). The dashed line signals that it's just a boundary, not part of the treasure itself.
Shade the Correct Region
Time to shade! We need to figure out which side of the line satisfies the inequality -x + 4y > 12. Let's use our trusty test point (0, 0) again:
-0 + 4(0) > 12 0 > 12
This is false! Since (0, 0) doesn't satisfy the inequality, we shade the region opposite to the one containing (0, 0). So, we shade the region above the dashed line. Great, we've graphed our second inequality!
Step 3: Identifying the Feasible Region
Here comes the exciting part: finding the solution to the system of inequalities. We've graphed each inequality separately, and now we need to put them together. Remember, the solution is the region where both inequalities are true at the same time. This is the overlapping region of the shaded areas from our individual graphs.
Overlaying the Graphs
Imagine placing the two graphs we've created on top of each other. You'll see four distinct regions: the region shaded by only the first inequality, the region shaded by only the second inequality, the region shaded by both inequalities, and the region shaded by neither. The solution to our system is the region shaded by both inequalities. It's like the area where the colors from two different maps overlap – that’s where the treasure is!
The Feasible Region
This overlapping region is often called the feasible region or the solution set. It represents all the (x, y) points that satisfy both inequalities simultaneously. In our case, it's the area above the dashed line (-x + 4y > 12) and below the solid line (x + 2y ≤ 6). This region might be bounded (enclosed by lines) or unbounded (extending infinitely in some direction). The shape and size of the feasible region give us valuable information about the possible solutions to our system.
Conclusion: Finding the Solution Set
So, what have we accomplished? We've successfully navigated the world of system of inequalities! We started with a system:
And we walked through the process of graphing each inequality, identifying the feasible region, and understanding that this region represents the solution set. Remember, the solution isn't a single point, but an entire area on the coordinate plane. This area contains all the (x, y) pairs that make both inequalities true.
Key Takeaways
Let's recap the key steps we've learned:
- Convert inequalities to equations to find the boundary lines.
- Find two points on each line to graph them.
- Determine if the line is solid or dashed based on the inequality symbol.
- Shade the correct region using a test point.
- Identify the overlapping region (feasible region) as the solution set.
Understanding system of inequalities is a valuable skill in mathematics, with applications in various fields like economics, computer science, and engineering. Keep practicing, and you'll become a pro at solving these problems! Keep exploring, keep learning, and most importantly, have fun with math! You guys got this!