These eight original questions follow the lessons in linear inequalities in two variables. They ramp in difficulty and each answer shows the reasoning.
Sketch each graph on paper, then open the answer.
The questions
Question 1. Find the intercepts of 3x + 2y = 12 and state whether the boundary of 3x + 2y < 12 is solid or dashed.
Answer
With x = 0, y = 6, giving (0, 6). With y = 0, x = 4, giving (4, 0). The sign is <, so the boundary is dashed.
Question 2. Which of these points satisfy 2x + y ≤ 8: (1, 5), (3, 3), (4, 0), (−1, 10)?
Answer
(1, 5): 2 + 5 = 7, true. (3, 3): 6 + 3 = 9, false.
(4, 0): 8, true, on the solid boundary.
(−1, 10): −2 + 10 = 8, true, also on the boundary. So (1, 5), (4, 0) and (−1, 10) satisfy it.
Question 3. Use a test point to decide the shaded side of x − y ≤ 3.
Answer
The line x − y = 3 passes through (0, −3) and (3, 0). Test (0, 0): 0 ≤ 3 is true, so shade the side containing the origin, which is above the line.
Question 4. Decide the shaded side of y > 2x. Is the boundary solid or dashed?
Answer
The boundary is dashed because of >. The line passes through the origin, so test (1, 0) instead: 0 > 2 is false. Shade the side without (1, 0), which is above and to the left of the line.
Question 5. Does (2, 3) satisfy y ≥ x + 1? Does it satisfy y > x + 1?
Answer
For y ≥ x + 1: 3 ≥ 3 is true, so it does. For y > x + 1: 3 > 3 is false, so it does not. The point is on the boundary, which belongs to the region only when the line is solid.
Question 6. Region R satisfies x ≥ 1, y ≥ 1 and x + y ≤ 5. Find its vertices and count the points with whole-number coordinates in R, including its edges.
Answer
Vertices: (1, 1), (4, 1) and (1, 4). Count by x: x = 1 gives y = 1 to 4 (4 points); x = 2 gives y = 1 to 3 (3); x = 3 gives y = 1 to 2 (2); x = 4 gives y = 1 (1). Total 4 + 3 + 2 + 1 = 10 points.
Question 7. Repeat Question 6 with x + y < 5. How many whole-number points are in R?
Answer
Points on x + y = 5 are now excluded, so x + y ≤ 4. By x: x = 1 gives y = 1 to 3 (3 points); x = 2 gives y = 1 to 2 (2); x = 3 gives y = 1 (1). Total 6 points.
Question 8. A solid line passes through (0, 4) and (4, 0), and the shaded region contains the origin. Write the inequality.
Answer
The line is x + y = 4. The origin gives 0 + 0 = 0, and 0 ≤ 4 is true, so the region is x + y ≤ 4. The line is solid, which matches ≤.
If you got these wrong
Questions 1 and 2 belong to drawing an inequality boundary correctly. Questions 3 to 5 belong to choosing the shaded side using a test point.
Questions 6 to 8 need finding a region satisfying several inequalities. Record slips in the mistake log tool, then try a timed round with the timed practice session builder.
If the same step keeps failing, SPM Mathematics one-to-one tuition lets a teacher trace it in your own working.