How do you know which half plane to shade?

Cracking the Code: Which Side Do You Shade? Graphing Linear Inequalities Made Easy

So, you’re tackling linear inequalities and graphing them… but which side do you actually shade? It’s a question that trips up a lot of people, so don’t worry if you’ve ever felt lost. Basically, instead of just drawing a line like with regular equations, we’re coloring in a whole area of the graph to show all the possible solutions. That area is called a half-plane.

Think of it like this: a line splits the graph in two, and we need to figure out which side holds all the answers to our inequality.

First things first: What is a Linear Inequality, Anyway?

You’ve probably seen these before: things like ax + by > c, or maybe y ≤ 2x + 1. See the “greater than” or “less than” symbols? That’s what makes it an inequality. Unlike an equation that has one specific answer, an inequality has a whole range of answers. Our job is to show that range on a graph.

Drawing the Line: Your Boundary

Before you can even think about shading, you’ve gotta draw the line itself. Just pretend the inequality sign is an equals sign and graph the line like normal. So, if you have y > 2x + 1, you’d graph the line y = 2x + 1. Easy peasy, right?

Now, here’s a little detail that makes a BIG difference:

• Solid or Dashed? If your inequality has that little “or equal to” bit (≤ or ≥), draw a solid line. That means the line itself is part of the solution. But if it’s just a plain old < or >, make it a dashed line. The line is a boundary, but not included.

The Shading Showdown: Picking a Side

Okay, this is the main event! How do we know which side to shade? I’ve got two tricks up my sleeve for you.

Trick #1: Slope-Intercept Sleuthing

This is my go-to when the inequality is in that nice y = mx + b format. It’s super simple:

• y > mx + b: Shade above the line. Think “greater,” think “higher up.”
• y ≥ mx + b: Same deal, shade above (and remember, it’s a solid line this time).
• y < mx + b: Shade below the line. “Less than” means “lower down.”
• y ≤ mx + b: Shade below (solid line again).

Important! If your inequality isn’t in slope-intercept form, you gotta rearrange it first. Get that y all by itself on one side. And here’s a pro tip: if you multiply or divide by a negative number while you’re rearranging, FLIP the inequality sign! Trust me, you’ll forget this at least once (we all do!), but try to remember it.

Trick #2: The Test Point Tango

This trick is foolproof, no matter what the inequality looks like.

Example Time! Let’s say we have 2x + y ≤ 4.

Horizontal and Vertical Line Quirks

Horizontal and vertical lines are a little special. Just remember:

• y > a: Shade above the horizontal line y = a.
• y < a: Shade below the horizontal line y = a.
• x > a: Shade to the right of the vertical line x = a.
• x < a: Shade to the left of the vertical line x = a.

Systems of Inequalities: The Overlap Zone

Now, if you’re dealing with multiple inequalities at once (a system of inequalities), you’re looking for the area where all the shadings overlap. It’s like a Venn diagram, but with shaded regions.

Wrapping It Up

Graphing linear inequalities might seem tricky at first, but with a little practice, you’ll get the hang of it. Just remember the solid vs. dashed line rule, pick your shading method, and double-check your work. Before you know it, you’ll be shading like a pro!