Graphing Inequalities: Y ≥ (4/5)x - (1/5) & Y ≤ 2x + 6

by ADMIN 55 views

Hey everyone! Today, we're diving into the world of graphing systems of inequalities. Specifically, we're going to tackle the question: How do you graph the system of inequalities y ≥ (4/5)x - (1/5) and y ≤ 2x + 6? Don't worry, it's not as scary as it looks! We'll break it down into easy-to-follow steps, so you'll be a pro in no time. Understanding how to visually represent these inequalities is super important in math, especially when you get into linear programming and optimization problems. So, let’s jump right in and make sure you've got a solid grasp of this concept. First off, we'll discuss the basics of linear inequalities and their graphical representation. Then, we will move on to the step-by-step process of graphing the given system. Finally, we’ll cover how to identify the solution set.

Understanding Linear Inequalities

Before we jump into our specific problem, let's make sure we're all on the same page about linear inequalities. A linear inequality is similar to a linear equation, but instead of an equals sign (=), it has an inequality sign (>, <, ≥, or ≤). This means that instead of a single line, we're dealing with a region on the graph. Let's talk about what each symbol means graphically:

  • y > mx + b: This represents all the points above the line y = mx + b. We use a dashed line to show that the points on the line are not included in the solution.
  • y < mx + b: This represents all the points below the line y = mx + b. Again, we use a dashed line because the points on the line are not part of the solution.
  • y ≥ mx + b: This represents all the points on or above the line y = mx + b. We use a solid line to show that the points on the line are included in the solution.
  • y ≤ mx + b: This represents all the points on or below the line y = mx + b. We use a solid line here too, indicating that the points on the line are part of the solution.

The m in mx + b represents the slope of the line, which tells us how steep the line is and in which direction it's going (uphill or downhill). The b represents the y-intercept, which is the point where the line crosses the y-axis. Knowing these basics, we can start plotting our lines accurately.

When we graph inequalities, we're essentially shading the area that satisfies the inequality. Think of it like this: every point in the shaded region, when plugged into the inequality, will make the statement true. This visual representation is super helpful because it allows us to see all the possible solutions at a glance. Sometimes, we'll have one inequality, and other times, we'll have a system of inequalities, like in our problem today. A system just means we have two or more inequalities that we need to satisfy at the same time. The solution to a system of inequalities is the region where the shaded areas of all the inequalities overlap. This overlapping region contains all the points that make all the inequalities true. So, with that foundation in place, we’re ready to tackle our specific problem and graph those inequalities!

Step-by-Step: Graphing y ≥ (4/5)x - (1/5)

Alright, let's get to the fun part: graphing! We'll start with the first inequality: y ≥ (4/5)x - (1/5). Remember, the goal here is to draw the line that represents the equation and then shade the correct region. So, grab your graph paper (or your favorite graphing app) and let’s get started. This first step is crucial because accurately plotting the lines will determine the correct solution region. If the line is off, the entire graph will be incorrect, so pay close attention to detail.

  1. Convert the Inequality to Slope-Intercept Form (if needed): In this case, our inequality is already in slope-intercept form (y ≥ mx + b), which makes our job easier. We can easily identify the slope and y-intercept. Our slope (m) is 4/5, and our y-intercept (b) is -1/5. If the inequality wasn't in this form, we'd need to rearrange it to isolate y on one side. This might involve adding, subtracting, multiplying, or dividing both sides of the inequality, being careful to flip the inequality sign if we multiply or divide by a negative number.
  2. Plot the y-intercept: The y-intercept is the point where the line crosses the y-axis. In our case, it's -1/5, which is a little less than 0. Plot this point on your graph. This is our starting point for drawing the line. Think of it as the anchor that the rest of the line will be based on. An accurate y-intercept ensures that the entire line is positioned correctly on the graph.
  3. Use the Slope to Find Another Point: The slope is “rise over run,” which means for every 5 units we move to the right (run), we move 4 units up (rise). Starting from our y-intercept (-1/5), move 5 units to the right and 4 units up. Plot this new point. Now you have two points that define your line. Using the slope to find a second point is a reliable method to ensure the line has the correct angle and direction. It's like having two landmarks to guide you in drawing a straight path.
  4. Draw the Line: Since our inequality is y ≥ (4/5)x - (1/5), it includes the