Solving Inequalities: Finding The Solution Set

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Hey guys! Ever found yourself staring at a bunch of inequalities and wondering what they all mean together? Well, you're in the right place! Today, we're going to break down how to find the solution set for a system of inequalities. Specifically, we'll tackle this problem: x ≥ 0; y ≥ 0; 2x + y ≤ 6; x + 2y ≤ 8. Trust me; it's not as scary as it looks. We’ll go through each step, making sure you understand the logic behind it. So grab your pencils, and let’s dive in!

Understanding the Inequalities

Before we start plotting lines and shading regions, let's make sure we understand what each inequality means. This is super important because each inequality represents a condition that our solution must satisfy.

  • x ≥ 0: This simply means that our solution must lie on the right side of the y-axis (including the y-axis itself). In other words, x-values must be non-negative.
  • y ≥ 0: Similarly, this means our solution must lie above the x-axis (including the x-axis). So, y-values must be non-negative.
  • 2x + y ≤ 6: This is a linear inequality. To understand it better, we can think of the equation 2x + y = 6. This is a straight line. The inequality means we're looking for the region below or on this line.
  • x + 2y ≤ 8: Just like the previous one, this is another linear inequality. Think of the line x + 2y = 8. The inequality means we want the region below or on this line.

Basically, the first two inequalities restrict us to the first quadrant (where both x and y are positive), while the last two give us specific boundaries within that quadrant. The solution set will be the region that satisfies all four inequalities simultaneously. Finding this region involves graphing each inequality and identifying the area where all shaded regions overlap. This overlapping area is the solution set—the set of all points (x, y) that make all the inequalities true at the same time.

Graphing the Inequalities

Okay, now comes the fun part: graphing! Graphing these inequalities helps us visualize the solution set. We'll take it step by step.

  1. Graph x ≥ 0 and y ≥ 0:

    • The inequality x ≥ 0 is represented by the region to the right of the y-axis. Shade this region lightly.
    • The inequality y ≥ 0 is represented by the region above the x-axis. Shade this region lightly as well.
    • Together, these two inequalities restrict our solution to the first quadrant.

  2. Graph 2x + y ≤ 6:

    • First, let's graph the line 2x + y = 6. To do this, find two points on the line.
    • If x = 0, then y = 6. So, (0, 6) is one point.
    • If y = 0, then 2x = 6, so x = 3. Thus, (3, 0) is another point.
    • Plot these two points and draw a line through them. Since the inequality is ≤, we draw a solid line (this indicates that the line itself is included in the solution).
    • Now, we need to determine which side of the line to shade. Pick a test point, like (0, 0), and plug it into the inequality: 2(0) + 0 ≤ 6. This simplifies to 0 ≤ 6, which is true. Therefore, we shade the region that includes the origin (0, 0).

  3. Graph x + 2y ≤ 8:

    • Similarly, let's graph the line x + 2y = 8.
    • If x = 0, then 2y = 8, so y = 4. The point (0, 4) is on the line.
    • If y = 0, then x = 8. The point (8, 0) is also on the line.
    • Plot these points and draw a solid line through them (again, because the inequality is ≤).
    • Pick a test point, like (0, 0), and plug it into the inequality: 0 + 2(0) ≤ 8. This simplifies to 0 ≤ 8, which is true. Shade the region that includes the origin (0, 0).

Identifying the Feasible Region

Alright, so now you should have a graph with all the inequalities plotted. The next step is to find the feasible region. The feasible region, also known as the solution set, is the area where all the shaded regions overlap. It's the region that satisfies all the inequalities simultaneously. It's like finding the common ground where all the conditions are met. This region is usually a polygon, and its vertices (corner points) are particularly important because they often represent the solutions in optimization problems.

  1. Look for Overlap: Identify the area on your graph where all the shaded regions overlap. This area will be bounded by the x-axis, the y-axis, and the two lines you graphed.
  2. Find the Vertices: The vertices of the feasible region are the points where the boundary lines intersect. These points are crucial for determining the exact solution set. We already know a few vertices: (0, 0), (3, 0), and (0, 4). We need to find the intersection point of the two lines 2x + y = 6 and x + 2y = 8.

Finding the Intersection Point

To find where the lines 2x + y = 6 and x + 2y = 8 intersect, we need to solve these equations simultaneously. There are a couple of ways to do this:

  1. Substitution Method:

    • Solve one equation for one variable. For example, from the first equation, we can express y as: y = 6 - 2x.
    • Substitute this expression into the second equation: x + 2(6 - 2x) = 8.
    • Simplify and solve for x: x + 12 - 4x = 8 → -3x = -4 → x = 4/3.
    • Now, substitute the value of x back into the equation y = 6 - 2x to find y: y = 6 - 2(4/3) = 6 - 8/3 = 10/3.
    • So, the intersection point is (4/3, 10/3).

  2. Elimination Method:

    • Multiply the equations by suitable constants so that the coefficients of one variable are equal but opposite in sign. For example, multiply the first equation by -2: -4x - 2y = -12. The second equation is x + 2y = 8.
    • Add the two equations to eliminate y: (-4x - 2y) + (x + 2y) = -12 + 8 → -3x = -4 → x = 4/3.
    • Substitute the value of x back into either of the original equations to find y. Using the first equation: 2(4/3) + y = 6 → 8/3 + y = 6 → y = 6 - 8/3 = 10/3.
    • Again, the intersection point is (4/3, 10/3).

Thus, we've found that the two lines intersect at the point (4/3, 10/3). This is another vertex of our feasible region. So now we know all vertices, (0,0), (3,0), (0,4) and (4/3, 10/3).

Defining the Solution Set

Now that we've identified the feasible region and its vertices, we can define the solution set. The solution set includes all points (x, y) within the feasible region, including the points on the boundary lines.

In this case, the solution set is the region bounded by the points (0, 0), (3, 0), (4/3, 10/3), and (0, 4). Any point within this region satisfies all the given inequalities: x ≥ 0, y ≥ 0, 2x + y ≤ 6, and x + 2y ≤ 8. To summarize:

  • Feasible Region: The area enclosed by the vertices (0, 0), (3, 0), (4/3, 10/3), and (0, 4).
  • Solution Set: All points (x, y) within this feasible region, including the boundary lines.
  • Vertices: (0, 0), (3, 0), (4/3, 10/3), and (0, 4)

Practical Applications and Tips

Understanding how to find the solution set for a system of inequalities isn't just an abstract math exercise. It has practical applications in various fields, such as:

  • Linear Programming: Used in business and economics to optimize resources subject to constraints. The feasible region represents the set of possible solutions, and the vertices often indicate optimal solutions.
  • Resource Allocation: Helps in deciding how to allocate resources (like time, money, or materials) efficiently, given certain limitations.
  • Engineering Design: Used to ensure that designs meet certain specifications and constraints.

Some tips for solving these problems:

  • Always double-check your graphs: Make sure you've shaded the correct regions for each inequality.
  • Use a test point: Picking a test point can help you verify which side of a line to shade.
  • Be careful with solid vs. dashed lines: Solid lines indicate that the boundary is included in the solution, while dashed lines mean it's not.

So there you have it! Finding the solution set for a system of inequalities might seem daunting at first, but with a step-by-step approach, it becomes much more manageable. Remember to understand the inequalities, graph them accurately, identify the feasible region, and define the solution set. Keep practicing, and you'll become a pro in no time!