How do you graph the inequality y 2x?

Graphing Linear Inequalities: No Sweat Guide to y ≤ 2x

So, you’re staring at y ≤ 2x and thinking, “How do I even graph that?” Don’t sweat it! Graphing linear inequalities isn’t as scary as it sounds. It’s really just taking what you know about graphing lines and adding a little shading to the mix. Instead of just one line, we’re painting a picture of a whole bunch of possible solutions on a coordinate plane. Think of it as finding not just one right answer, but a whole neighborhood of them.

Cracking the Code: What are Linear Inequalities?

Okay, quick recap. A linear inequality is like a regular equation, but instead of an equals sign, you’ve got one of these: <, >, ≤, or ≥. In our case, y ≤ 2x is saying, “Find all the points (x, y) where the y-value is either less than or equal to two times the x-value.” In other words, we’re not just looking for points on a line, but everything below it as well.

Step 1: Draw the Line (the Boundary, that is)

First things first, let’s pretend that inequality is an equality. Let’s graph the line y = 2x. This line is super important; it’s the boundary. It’s what separates the “yes” region from the “no” region on our graph.

• Slope-intercept deconstructed: Remember y = mx + b? That’s slope-intercept form, and it’s our friend. ‘m’ is the slope (how steep the line is), and ‘b’ is the y-intercept (where the line crosses the y-axis). For y = 2x, the slope (m) is 2 (or 2/1, which means up two, over one), and the y-intercept (b) is 0. It crosses the y axis at the origin.
• Plotting Time: Start at the y-intercept (0, 0). Now, use that slope! Go up 2 units on the y axis, and then move one unit to the right on the x axis. Boom! You’ve got another point. Connect the dots, and you’ve got your line.
• Solid or Dashed? A Crucial Choice: Here’s where the inequality symbol matters. Because we have “less than or equal to” (≤), our boundary line is solid. A solid line means that all the points on the line are also solutions. They’re part of the club! If it was just y < 2x, we'd use a dashed line. A dashed line is like saying, "You can get really close, but you can’t actually be on this line.”

Step 2: Shady Business – Finding the Right Region

That boundary line just chopped our coordinate plane in half. One side is the solution zone; the other is a wasteland of wrong answers. How do we know which side to shade? Test point time!

• Pick a Test Point: Grab any point that’s not on the line. (0,0) is usually easiest, but since our line goes right through it, let’s pick something else. How about (1, 1)?

• The Moment of Truth: Testing the Point: Plug the x and y values of (1, 1) into the original inequality:

  1 ≤ 2(1)
  
  
  1 ≤ 2

• The Verdict: Is that true? Yep! 1 is definitely less than or equal to 2. Since the inequality is true with our test point, we shade the region that includes (1, 1). If it was false, we’d shade the other side.

Step 3: Shade Away!

Grab your pencil (or your stylus, whatever) and shade the heck out of the region that contains your test point. That shaded area, plus the solid boundary line, is the visual representation of all the solutions to y ≤ 2x. You’ve just mapped out all the points where the y-value is less than or equal to twice the x-value.

In a Nutshell

Graphing y ≤ 2x is all about drawing the line y = 2x (solid, because of the “or equal to” part) and then shading everything below it. You’re showing all the possible solutions to the inequality, not just one single line. Easy peasy, right?