y < 2x + 1 is satisfied by a whole half of the plane, not by a single line.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The boundary
Graph the matching equation first. That line separates what works from what does not.
Solid or dashed
Use a solid line for ≤ or ≥, because the line itself belongs. Use dashed for < or >.
Which side to shade
Test one point not on the line — the origin is easiest. If it works, shade its side.
The solution is a region
y > 2x + 1 is satisfied by every point above the line y = 2x + 1. With two variables the solution set is an area of the plane, not a stretch of a line.
Solid or dashed boundary
Draw the boundary line dashed for strict inequalities and solid for ≤ or ≥. The line style records whether points on the boundary are included, which is information the shading alone cannot convey.
Test a point to choose the side
Pick a point not on the line — the origin is ideal when it is not on the boundary — and check whether it satisfies the inequality. If it does, shade its side. Testing beats trying to remember which direction each symbol shades.
Several inequalities together
A system of inequalities is satisfied only where all the shaded regions overlap. That overlap is the feasible region, and finding it is the foundation of linear programming — how businesses optimise under constraints.
Step 2: Try It Yourself
The boundary line. The solutions lie entirely on one side of it.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Priya Test the Origin
Priya graphs y < 2x + 1.
- Step 1
She draws y = 2x + 1 as the boundary.
Step 4: Your Turn
Practice makes it stick.
Testing a Point
Problem 1 of 2
Does (0, 0) satisfy y < 2x + 1? 1 for yes, 0 for no.
The Line Style
Problem 2 of 2
For y ≤ 3x, is the boundary solid? 1 for yes, 0 for no.
Boundary and Side
1 of 8
y > x + 2. Is the boundary dashed? 1 for yes, 0 for no.
2 of 8
Does (0, 0) satisfy y > x + 2? 1 for yes, 0 for no.
3 of 8
Does (0, 5) satisfy y > x + 2? 1 for yes, 0 for no.
4 of 8
y ≥ 2x. Is the boundary solid? 1 for yes, 0 for no.
5 of 8
Order the steps for graphing a linear inequality.
- 1Make it solid or dashed
- 2Test a point off the line
- 3Shade the side that works
- 4Graph the boundary line
6 of 8
Does (2, 1) satisfy y < x? 1 for yes, 0 for no.
7 of 8
Can the origin be used as a test point for y < x? 1 for yes, 0 for no.
8 of 8
How many points satisfy a linear inequality in two variables? Use 99 for infinitely many.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Does (0, 0) satisfy y < x − 3? 1 for yes, 0 for no.
What You Learned
- A linear inequality in two variables is satisfied by half the plane.
- The boundary is solid when equality is included and dashed when it is not.
- Testing one point off the line decides which side to shade.