Graphing Inequalities: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the fascinating world of graphing inequalities. Specifically, we'll tackle the inequality y ≤ 3x - 6. Don't worry, it's not as intimidating as it sounds! We'll break it down step-by-step, making it super easy to understand. By the end of this guide, you'll be graphing inequalities like a pro. So, grab your graph paper (or your favorite digital graphing tool), and let's get started!

Understanding Linear Inequalities

Before we jump into graphing, let's quickly recap what linear inequalities are. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike linear equations, which have a single solution (or a few), linear inequalities have a range of solutions. Think of it as a whole bunch of points on a graph that satisfy the inequality. These points together form a region, and that's what we'll be graphing today.

In our case, we have y ≤ 3x - 6. This inequality tells us that we're looking for all the points (x, y) where the y-value is less than or equal to 3 times the x-value, minus 6. Sounds a bit complicated, right? But trust me, the graphing process will make it crystal clear. The main thing to remember is that inequalities represent a range of possible solutions, not just a single point. This range is visually represented as a shaded region on the coordinate plane. So, we're not just drawing a line; we're coloring in an area that includes all the points that make our inequality true. Now that we have a basic understanding of linear inequalities, let's move on to the actual steps of graphing.

Step 1: Treat the Inequality as an Equation

The first step in graphing an inequality is to treat it as if it were a regular equation. This means we replace the inequality symbol (in our case, ≤) with an equals sign (=). So, y ≤ 3x - 6 becomes y = 3x - 6. Why do we do this? Because the equation y = 3x - 6 represents a straight line, and this line will be the boundary of our shaded region. It's like drawing the fence that separates the solutions from the non-solutions. Think of this line as the backbone of our graph. It's the reference point from which we'll determine which side of the plane to shade. So, before we can shade anything, we need to draw this line accurately.

Now, let's think about how to graph y = 3x - 6. There are a couple of ways to do this. One common method is to find the slope and y-intercept. In the slope-intercept form (y = mx + b), the slope (m) is 3, and the y-intercept (b) is -6. This means the line crosses the y-axis at the point (0, -6), and for every 1 unit we move to the right, the line goes up 3 units. Another method is to find two points that lie on the line. We can do this by choosing any two x-values, plugging them into the equation, and solving for y. For example, if we let x = 0, we get y = 3(0) - 6 = -6, giving us the point (0, -6). If we let x = 2, we get y = 3(2) - 6 = 0, giving us the point (2, 0). Once we have these two points, we can simply draw a line through them. No matter which method you choose, the goal is to accurately represent the line y = 3x - 6 on your graph. This line is crucial because it defines the boundary of the solutions to our inequality.

Step 2: Determine the Type of Line

Okay, we've got our equation, y = 3x - 6, graphed as a line. But here's a crucial question: Should this line be solid or dashed? The answer lies in the original inequality symbol. If the inequality includes an