A linear inequality in x and y splits the graph into two sides, and the boundary is the line between them. Draw the line for the equals version, use solid or dashed to match the symbol, then test one point to choose the side.
This follows representing an interval on a number line, where open and closed circles did the same job in one dimension. It leads to identifying a feasible region, where several boundaries combine.
How do you draw the boundary?
- Replace the inequality symbol with = to get the boundary equation.
- Find two or three points on that line. Intercepts are quick: set x = 0 for the y-intercept, and y = 0 for the x-intercept.
- Draw the line. Make it solid for ≤ or ≥ and dashed for < or >.
- Test a point to decide the side.
The bounds and rounding explainer shows the same idea of included and excluded end-points in a measurement setting.
Worked example 1: y > 2x − 3
Step 1, boundary: y = 2x − 3. Points: x = 0 gives y = −3, and x = 2 gives y = 1. So the line passes through (0, −3) and (2, 1).
Step 2, style: the symbol is >, which is strict, so the line is dashed.
Step 3, test (0, 0): is 0 > 2(0) − 3? That is 0 > −3, which is true. So the side containing the origin is the solution side, which is the region above the line.
Step 4, second check: try (3, 0). Is 0 > 6 − 3 = 3? No. (3, 0) lies below the line, which is not shaded. The two checks agree.
Worked example 2: 2x + 3y ≤ 12
Step 1, boundary: 2x + 3y = 12. When x = 0, 3y = 12, so y = 4. When y = 0, 2x = 12, so x = 6. The line joins (0, 4) and (6, 0).
Step 2, style: the symbol is ≤, so the line is solid.
Step 3, test (0, 0): 0 ≤ 12 is true, so shade the side containing the origin, which is below and to the left of the line.
The mistake to watch for
Mistaken working: 2x − y < 4 has the symbol ”<”, so shade below the line.
The student used “less than means below” without testing a point.
That rule works only when the inequality is written as y < something. Here the y term is negative.
Test (0, 0): 2(0) − 0 = 0, and 0 < 4 is true, so the origin side is the solution. The boundary 2x − y = 4 passes through (2, 0) and (0, −4), and the origin is above this line.
So the correct region is above the line, the opposite of the student’s answer.
If you rearrange to y > 2x − 4, the symbol turns round because you divided by −1, which links back to reversing an inequality. The test point avoids all that.
Check yourself
1. Is the boundary for x + y < 5 solid or dashed?
Show answer
The symbol is <, so points on the line do not satisfy it. The line is dashed.
2. Find the intercepts of the boundary 3x + 2y = 12.
Show answer
Set x = 0: 2y = 12, so y = 6. Set y = 0: 3x = 12, so x = 4.
(4, 0) and (0, 6)
3. Describe the region y ≥ x − 1.
Show answer
Boundary y = x − 1 passes through (0, −1) and (1, 0). The symbol is ≥, so the line is solid. Test (0, 0): 0 ≥ −1 is true, so shade the side with the origin.
Solid line, shade above the line
Where this leads next
Once one boundary is secure, move to identifying a feasible region from combined constraints. The non-calculator working trainer helps with the intercept arithmetic.
If test points feel slow, a teacher in online one-to-one Mathematics tuition can practise the routine with you on fresh inequalities until it becomes quick.