Unraveling Inequalities: Exploring y = mx + b

Welcome back! This week, we’re diving into the world of linear inequalities and exploring how they manifest in the familiar equation y = mx + b. Get ready to embark on a journey through greater than, less than, greater than or equal to, and less than or equal to, as we unravel the mysteries of graphing linear inequalities.

Understanding Linear Inequalities

Before we jump into graphing, let’s brush up on our understanding of linear inequalities. In the equation y = mx + b, ‘m’ represents the slope of the line, and ‘b’ represents the y-intercept. When we introduce inequalities, we’re essentially defining regions on the graph where certain conditions are met.

Graphing Inequalities

Graphing linear inequalities is like shading regions on a map to indicate where certain conditions hold true. Here’s how it works:

1. Greater Than (>): If we have y > mx + b, we shade the region above the line. When
2. Less Than (<): Conversely, for y < mx + b, we shade the region below the line.
3. Greater Than or Equal To (≥): For y ≥ mx + b, we include the line in the shading, indicating that points on the line or above it.
4. Less Than or Equal To (≤): Similarly, for y ≤ mx + b, we include the line in the shading, indicating that points on the line or below it.

NOTE- To figure out which region you have to shade, we can use (0,0) points in the formula. If when you solve it becomes true then that means you shade that region, and if it doesn’t you shade the opposite region.

NOTE- When ever we have >,< without the equal sign that means we have a dotted line. And whenever we have ≥,≤ that makes just the line without the dots.

Examples

Let’s explore some examples to solidify our understanding:

1. y > 2x + 1:
   • We start by graphing the line y = 2x + 1.
   • 
     • Next, we shade the region above the line to indicate all points where y is greater than 2x + 1.
2. y < -3x + 2:
   • We graph the line y = -3x + 2.
   • 
   • Then, we shade the region below the line to indicate all points where y is less than -3x + 2.
3. y ≥ x – 3:
   • We graph the line y = x – 3.
   • 
   • We shade the region above the line and include the line itself to show that all points on or above the line satisfy the inequality.
4. y ≤ 4x + 5:
   • We graph the line y = 4x + 5.
   • 
   • We shade the region below the line and include the line itself to indicate that all points on or below the line.

Conclusion

By graphing linear inequalities in the form y = mx + b, we gain valuable insights into the relationships between variables and conditions. Whether it’s shading regions above, below, or including the lines themselves, graphing inequalities allows us to visualize solutions and make informed decisions in various mathematical contexts. So, let’s continue to explore the fascinating world of inequalities and unleash the power of graphing to illuminate our mathematical journey!