Solving 2x + Y > 4: A Step-by-Step Guide

by TextBrain Team 41 views

Hey guys! Today, we're diving into a fun little math problem: solving the inequality 2x + y > 4, with the added twist that x > 0 and y > 0. Sounds like a party, right? Let's break it down and make it super easy to understand. We'll go through each step, explain why we're doing it, and by the end, you'll be a pro at solving these types of inequalities. So, grab your pencils, and let's get started!

Understanding the Basics

Before we jump into the nitty-gritty, let's make sure we're all on the same page with the basics. An inequality, like 2x + y > 4, is a mathematical statement that shows a relationship between two expressions that are not necessarily equal. In this case, we're saying that the expression 2x + y is greater than 4. This is different from an equation (like 2x + y = 4), where we're looking for specific values that make the two sides equal.

The constraints x > 0 and y > 0 are also important. They tell us that we're only interested in solutions where both x and y are positive numbers. This limits the possible solutions and makes the problem a bit more manageable. Think of it like setting boundaries for a game – we know where we can and can't play.

Graphing Linear Inequalities

Graphing linear inequalities is a visual way to represent all the possible solutions. When we graph an inequality like 2x + y > 4, we're essentially drawing a line on a coordinate plane that divides the plane into two regions. One region contains all the points (x, y) that satisfy the inequality, and the other region contains all the points that don't. The line itself is called the boundary line.

To graph the boundary line, we first treat the inequality as an equation and solve for y. In this case, we rewrite 2x + y > 4 as y > -2x + 4. The equation of the boundary line is then y = -2x + 4. We can graph this line by finding two points on the line and drawing a straight line through them. For example, if we set x = 0, we get y = 4, so the point (0, 4) is on the line. If we set y = 0, we get x = 2, so the point (2, 0) is also on the line. Now, we can draw the line through these two points.

Since our inequality is y > -2x + 4 (greater than), we need to decide which side of the line represents the solution. To do this, we can pick a test point that is not on the line and plug its coordinates into the original inequality. If the inequality is true for that point, then the region containing that point is the solution region. If the inequality is false, then the other region is the solution region. A common test point is (0, 0). Plugging this into 2x + y > 4, we get 2(0) + 0 > 4, which simplifies to 0 > 4. This is false, so the region that does not contain (0, 0) is the solution region. We typically shade this region to indicate that it represents all the solutions to the inequality. Because the inequality is strict (greater than, not greater than or equal to), the boundary line is drawn as a dashed line to show that points on the line are not included in the solution.

Considering the Constraints

Now, let's consider our constraints: x > 0 and y > 0. These constraints tell us that we're only interested in the solutions that lie in the first quadrant of the coordinate plane (where both x and y are positive). This means we need to restrict our attention to the portion of the shaded region that falls within the first quadrant. The region where x > 0 is the area to the right of the y-axis, and the region where y > 0 is the area above the x-axis. Therefore, the solution to our inequality with the given constraints is the area in the first quadrant that is above the line y = -2x + 4.

So, when you put it all together, you're looking at a graph where you've drawn the line y = -2x + 4 (as a dashed line), shaded the region above the line, and then only kept the part of that shaded region that's in the top-right corner of the graph (the first quadrant). That area represents all the possible combinations of positive x and y values that make the inequality 2x + y > 4 true. Congrats, you've visually represented the solution!

Step-by-Step Solution

Okay, let's break down how to actually solve this inequality step-by-step. It’s not as scary as it looks, I promise!

  1. Rewrite the Inequality:
    • Start with our inequality: 2x + y > 4
    • We want to isolate y to make it easier to graph and understand. So, subtract 2x from both sides: y > -2x + 4
  2. Consider the Constraints:
    • We know that x > 0 and y > 0. This means we're only looking for solutions where both x and y are positive. This is super important because it limits the area where our solutions can be found. Think of it as drawing a box around our possible answers – we're only interested in what's inside the box!
  3. Graph the Boundary Line:
    • To graph the inequality, we first graph the boundary line. This is the line where y = -2x + 4.
    • Find two points on the line. A simple way to do this is to set x = 0 and solve for y, and then set y = 0 and solve for x.
      • If x = 0, then y = -2(0) + 4 = 4. So, the point (0, 4) is on the line.
      • If y = 0, then 0 = -2x + 4. Solving for x, we get 2x = 4, so x = 2. The point (2, 0) is on the line.
    • Plot these two points (0, 4) and (2, 0) on a graph and draw a dashed line through them. It's dashed because the inequality is > (greater than), not >= (greater than or equal to), which means points on the line are not included in the solution. If it were >=, we'd draw a solid line.
  4. Shade the Correct Region:
    • Now, we need to figure out which side of the line to shade. To do this, pick a test point that is not on the line. The easiest one is usually (0, 0).
    • Plug the test point into the original inequality: 2x + y > 4 becomes 2(0) + 0 > 4, which simplifies to 0 > 4.
    • Is 0 > 4? No, it's not! This means that the point (0, 0) is NOT part of the solution. So, we need to shade the region on the other side of the line from (0, 0). Shade the region above the dashed line. Make sure you only shade the portion of the region that's in the first quadrant (where x > 0 and y > 0).
  5. Identify the Solution Region:
    • The solution to the inequality 2x + y > 4 with the constraints x > 0 and y > 0 is the shaded region in the first quadrant above the dashed line y = -2x + 4. This region represents all the possible combinations of positive x and y values that make the inequality true.

And there you have it! You've successfully solved the inequality and identified the solution region. Give yourself a pat on the back; you've earned it!

Practical Examples

So, you might be thinking,