How do you plot an inequality?

Find the Hotspot (Critical Value): What’s the magic number where things change? In x > 3, it’s obviously 3. This is your critical value; the point where the inequality pivots.

Circle Time: Put a circle on that number line, right at your critical value. Now, pay attention – this is where it gets a little nuanced.

• Open Circle = Exclusive Club: If your inequality is a “less than” (<) or "greater than" (>), draw an open circle. Think of it as saying, “Hey, we’re getting close to 3, but 3 itself isn’t invited to the party.”
• Closed Circle = Everyone’s Welcome: If you’ve got a “less than or equal to” (≤) or “greater than or equal to” (≥), fill that circle in! This means, “Yep, 3 is part of the solution. Come on in!”

Shade the Truth: Now, grab your mental highlighter and shade the part of the number line that makes the inequality true.

• “Greater Than” Means “Go Right”: If it’s a > or ≥ situation, shade to the right. Those numbers are bigger, after all.
• “Less Than” Means “Head Left”: For < or ≤, shade to the left. Smaller numbers live that way.

Pretend It’s an Equation (For Now): Swap that inequality sign for an equals sign and graph the line. This line is your boundary, the edge of the solution zone.

Solid or Dashed? The Line’s Personality: This is crucial. Does the line get to join the party, or is it just a spectator?

• Solid Line = Inclusive: If your inequality includes “or equal to” (≤ or ≥), draw a solid line. The points on the line are part of the solution.
• Dashed Line = Exclusive: If it’s just < or >, make it a dashed line. Those points on the line? They’re not included.

Pick a Test Point (Be Smart About It): Choose a point that’s not on the line. (0, 0) is usually your best friend here, unless the line goes right through it.

Test Drive: Plug the x and y values of your test point into the original inequality.

Shade the Winner:

• Test Point Works? Shade That Side: If the test point makes the inequality true, shade the side of the line where that point lives. That’s your solution zone!
• Test Point Fails? Shade the Other Side: If the test point makes the inequality false, shade the other side. The solutions are hiding over there.