Graphing Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of inequalities and learning how to visualize them. Specifically, we'll be tackling how to graph the system of inequalities: $y \geq \frac{4}{5}x - \frac{1}{5}$ and $y \leq 2x + 6$. Don't worry, it's not as scary as it sounds! By the end of this guide, you'll be a pro at sketching these graphs and understanding the solution set. Let's get started, shall we?

Understanding Inequalities: The Basics

Before we jump into the specific inequalities, let's brush up on the fundamentals. An inequality, unlike an equation, doesn't state that two expressions are equal. Instead, it shows a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another. The symbols we use are: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). The solution to an inequality is not a single point, but a range of values. When graphing these, this translates to shading an entire region of the coordinate plane. Think of it like this: the inequality is saying, "Hey, the solution is somewhere in this area!"

To graph an inequality, we first need to graph the corresponding equation. This equation is the same as the inequality, but with the inequality sign replaced by an equals sign. For example, the corresponding equation for $y \geq \frac{4}{5}x - \frac{1}{5}$ is $y = \frac{4}{5}x - \frac{1}{5}$. The graph of the equation will be a line, and this line acts as the boundary between the solution region and the region that is not part of the solution. If the inequality includes "equal to" (${\geq}$ or ${\leq}$), then the boundary line is included in the solution, and we draw a solid line. If the inequality does not include "equal to" (${>}$ or ${<}$), the boundary line is not included, and we draw a dashed line. After drawing the boundary line, the next step is to determine which side of the line to shade. This is where we identify the region that contains all the points that satisfy the inequality. There are a couple of ways to do this, which we will explore below. Don't worry, we'll go through it step by step, making it easy to understand.

Step-by-Step Guide to Graphing $y \geq \frac{4}{5}x - \frac{1}{5}$

Alright, let's get down to business and graph the first inequality, $y \geq \frac{4}{5}x - \frac{1}{5}$. This inequality tells us that the y-values are greater than or equal to the expression $\frac{4}{5}x - \frac{1}{5}$.

1. Graph the Boundary Line: First, we need to graph the corresponding equation, which is $y = \frac{4}{5}x - \frac{1}{5}$. This is a linear equation in slope-intercept form ($y = mx + b$), where m is the slope and b is the y-intercept. In this case, the slope m is $\frac{4}{5}$, and the y-intercept b is $-\frac{1}{5}$. So, our line crosses the y-axis at $-\frac{1}{5}$. To plot the line, we can use the y-intercept and the slope. Starting from the y-intercept, we can move up 4 units and to the right 5 units (because the slope is 4/5). This gives us another point on the line. Since the inequality includes "equal to" (≥), we draw a solid line. This signifies that the line itself is part of the solution.

2. Determine the Shaded Region: Now comes the fun part: figuring out which side of the line to shade. There are two common methods to achieve this: testing a point or understanding the slope. Let's use the testing a point method. We pick a point that is not on the line. The easiest one to use is usually (0, 0), as long as the line doesn't go through the origin. Substitute x = 0 and y = 0 into the original inequality $y \geq \frac{4}{5}x - \frac{1}{5}$: $0 \geq \frac{4}{5}(0) - \frac{1}{5}$. This simplifies to $0 \geq -\frac{1}{5}$. Since this statement is true, the point (0, 0) is part of the solution set. Therefore, we shade the region above the line because the inequality requires that y-values are greater than or equal to the expression on the right side of the inequality. That includes the points above the line.

3. Visualization: The shaded region represents all the points $(x, y)$ that satisfy the inequality $y \geq \frac{4}{5}x - \frac{1}{5}$. Any point in this region, when plugged into the inequality, will make the statement true. Congratulations, you've graphed your first inequality!

Graphing $y \leq 2x + 6$: Another Example

Now, let's move on to the second inequality: $y \leq 2x + 6$. This one says that y-values are less than or equal to $2x + 6$.

1. Graph the Boundary Line: Start by graphing the corresponding equation: $y = 2x + 6$. This equation is also in slope-intercept form. The slope (m) is 2, and the y-intercept (b) is 6. So, the line crosses the y-axis at 6. The slope of 2 means that from any point on the line, we can go up 2 units and right 1 unit to find another point. Since the inequality includes "equal to" (≤), we draw a solid line.

2. Determine the Shaded Region: Again, we can use the test point method. Let's test the point (0, 0) again. Substitute x = 0 and y = 0 into the original inequality $y \leq 2x + 6$: $0 \leq 2(0) + 6$. This simplifies to $0 \leq 6$. Since this is true, the point (0, 0) is part of the solution set. Therefore, we shade the region below the line. Since the inequality requires that y-values are less than or equal to the expression on the right side of the inequality. That includes the points below the line.

3. Visualization: The shaded region represents all the points $(x, y)$ that satisfy the inequality $y \leq 2x + 6$.

Finding the Solution to the System of Inequalities: The Intersection

Now we've graphed both inequalities individually. The solution to the system of inequalities is the set of points that satisfy both inequalities simultaneously. This means we are looking for the region where the shaded areas of the two graphs overlap. Where the regions intersect, that's where the solution to the system lies!

To find the overlapping region, imagine overlaying the two graphs. You'll see a region where both shaded areas are present. This overlapping region represents the solution to the system. Any point in this region will satisfy both $y \geq \frac{4}{5}x - \frac{1}{5}$ and $y \leq 2x + 6$. The intersection of the two solution sets is itself a solution set, so any point within the intersecting shaded region is part of the solution. The solution is also bounded by the two lines, and the lines are included in the solution because both inequalities have the