A linear inequality divides the plane in two. Every point on one side satisfies it.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The boundary line
Graph the matching equation first. A solid line includes the boundary, for ≤ and ≥; a dashed line excludes it.
Which side to shade
Test a point not on the line, usually the origin. If it satisfies the inequality, shade its side.
A system
The solution to a system is where all the shaded regions overlap. A point must satisfy every inequality.
It can be empty
If the regions never overlap, the system has no solution.
Constraints in context
Real problems add x ≥ 0 and y ≥ 0, since quantities are rarely negative. Those count as inequalities too.
The solution is a region
A system of inequalities is satisfied where all the shaded regions overlap. That overlap is the feasible region, and every point in it satisfies every constraint simultaneously.
Solid or dashed
Draw the boundary dashed for strict inequalities and solid for those including equality. The line style records whether boundary points count, which the shading alone cannot show.
Test a point to choose the side
Substitute a point not on the boundary — the origin when possible — and shade its side if it satisfies the inequality. Testing is more reliable than remembering which symbol shades which way.
Where feasible regions matter
Budget limits, resource constraints and production capacities all become inequalities. Finding the best point in the feasible region is linear programming, which is how many scheduling and allocation problems are solved.
Step 2: Try It Yourself
Tap and try it out.
The slope is -1: for every 1 across, the line goes 1 down.
Step 3: Watch an Example
One step at a time.
Watch Rosa Shade a Region
Rosa graphs y < 2x + 1.
- Step 1
She draws y = 2x + 1 as the boundary.
Step 4: Your Turn
Practice makes it stick.
The Test
Problem 1 of 2
Does the origin satisfy y < 2x + 1? 1 yes, 0 no.
The Line
Problem 2 of 2
For y ≥ 3x − 2, is the boundary line solid or dashed? 1 solid, 2 dashed.
Shade the Region
1 of 8
For y > x + 1, solid or dashed boundary? 1 solid, 2 dashed.
2 of 8
For y ≤ 4x, solid or dashed? 1 or 2.
3 of 8
Does (0, 0) satisfy y > x + 3? 1 yes, 0 no.
4 of 8
Does (0, 0) satisfy y < 5 − x? 1 yes, 0 no.
5 of 8
Does (2, 3) satisfy y ≥ x? 1 yes, 0 no.
6 of 8
How many inequalities must a solution point satisfy in a system of three?
7 of 8
Put the graphing steps in order.
- 1Make it solid or dashed based on the symbol.
- 2Test a point not on the line.
- 3Shade the side that satisfies the inequality.
- 4Graph the boundary line.
8 of 8
Does (1, 1) satisfy y < 2x? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Does (0, 0) satisfy y ≤ 3x − 4? 1 yes, 0 no.
Question 2 of 2
What is the solution to a system of inequalities?
What You Learned
- An inequality shades half the plane on one side of a boundary line.
- Solid lines include the boundary; dashed lines exclude it.
- A system is solved by the region where every shading overlaps.