The boundary of a linear inequality is the line you get by changing the inequality sign to an equals sign. Draw it solid for ≤ and ≥, and dashed for < and >.
This lesson is part of linear inequalities in two variables within SPM Mathematics.
How do I find the line?
Take the inequality, swap the sign for =, and find two points. The intercepts are quickest: set x = 0 for the y-intercept, then y = 0 for the x-intercept.
For 2x + y ≤ 8, the equation is 2x + y = 8. With x = 0, y = 8, giving (0, 8). With y = 0, 2x = 8, so x = 4, giving (4, 0). Join them with a solid line because the sign is ≤.
Worked example: the same line, two signs
An original pair: draw the boundary for x + 2y < 6 and for x + 2y ≤ 6.
Both use the line x + 2y = 6. With x = 0, y = 3, giving (0, 3). With y = 0, x = 6, giving (6, 0).
For x + 2y < 6 draw the line dashed. For x + 2y ≤ 6 draw it solid. Test the point (6, 0) on the line: 6 + 0 = 6. It fails “< 6” and satisfies “≤ 6”, which is exactly why the line types differ.
What about horizontal and vertical lines?
An inequality with one variable also has a boundary. y ≥ 2 gives a horizontal solid line through (0, 2). x < −1 gives a vertical dashed line through (−1, 0).
Because the other variable is missing, any value of it works. That is why the line is flat or upright.
The mistake that costs marks
The common slip is drawing every boundary solid, because solid lines look neat. A strict inequality such as y > x + 1 needs a dashed line, and a solid line suggests points on the line are included.
The second slip is a wrong intercept. For 3x + 2y = 12, a student sets x = 0 and writes 3 × 0 + 2y = 12, then y = 12. The correct step is 2y = 12, so y = 6, giving (0, 6).
A self-check question
Draw the boundary of 3x + 2y > 12. State the intercepts and the line type.
Answer
Equation: 3x + 2y = 12. With x = 0, 2y = 12, so y = 6, giving (0, 6). With y = 0, 3x = 12, so x = 4, giving (4, 0). The sign is >, so the line is dashed.
Next, learn which side to shade in choosing the shaded side using a test point. The algebra step repair trainer drills the intercept substitution.
If your lines keep landing in the wrong place, SPM Mathematics one-to-one tuition lets a teacher check each intercept as you work.