Pick a point not on the line, put its coordinates into the inequality, and see if the statement is true. If it is, shade the side with that point. If not, shade the other side.
This lesson is part of linear inequalities in two variables. Draw the boundary first, as in drawing an inequality boundary correctly.
How does the test point work?
Take x + 2y ≤ 6, where the line x + 2y = 6 is solid. Use (0, 0): 0 + 0 = 0, and 0 ≤ 6 is true. So shade the side of the line that contains the origin.
For 3x + 2y ≥ 12, use (0, 0): 0 ≥ 12 is false. So shade the side that does not contain the origin.
Worked example: when the rule breaks
An original inequality: x − y ≤ 3. The boundary is x − y = 3, through (0, −3) and (3, 0), and it is solid.
Some students shade “below” because of the ≤ sign. Test (0, 0): 0 − 0 = 0, and 0 ≤ 3 is true, so shade the side with the origin. The origin is above the line, so the region is above.
Rewrite to confirm: −y ≤ 3 − x gives y ≥ x − 3, which means above the line. The negative sign on y reversed the direction.
A line through the origin
Take y ≥ 2x. The line y = 2x passes through (0, 0), so the origin cannot be used. Test (1, 0): 0 ≥ 2 is false.
So shade the side that does not contain (1, 0). The point (1, 0) lies below the line, so the region is above and to the left.
The mistake that costs marks
The usual slip is testing a point that lies on the line. For 2x + y ≤ 8, the point (4, 0) gives 8 ≤ 8, which is true on the line and tells you nothing about the sides. Always choose a point clearly off the line.
The other slip is testing correctly but shading the wrong side, for instance shading away from (0, 0) although the test was true. Write “true, so shade the side with (0, 0)” next to the graph before you shade.
A self-check question
Shade the region for 3x + 2y ≥ 12 using a test point. Then decide whether (5, 2) is in the region.
Answer
The line 3x + 2y = 12 is solid, through (0, 6) and (4, 0). Test (0, 0): 0 ≥ 12 is false, so shade the side away from the origin. Check (5, 2): 15 + 4 = 19, and 19 ≥ 12 is true, so (5, 2) is in the region.
Continue with finding a region satisfying several inequalities, where each line gets its own test. The algebra step repair trainer gives extra substitution drills.
If shading still feels like a guess, SPM Mathematics one-to-one tuition lets a teacher watch your test-point step in real time.