What you'll learn
Linear inequalities in two variables extend your knowledge of straight-line graphs to represent solution regions on the Cartesian plane. CXC CSEC Mathematics exam papers consistently test your ability to graph boundary lines, shade correct regions, and identify feasible solutions—typically worth 6-10 marks across multiple questions in Paper 2.
Key terms and definitions
Linear inequality — A mathematical statement comparing two expressions using inequality symbols (<, >, ≤, ≥) where the highest power of any variable is 1, such as 3x + 2y < 12.
Boundary line — The straight line obtained by replacing the inequality symbol with an equals sign; it forms the edge of the solution region.
Solution region (feasible region) — The area of the Cartesian plane containing all coordinate pairs (x, y) that satisfy the inequality or system of inequalities.
Solid line — Used for inequalities with ≤ or ≥ symbols, indicating points on the line are included in the solution.
Dashed (broken) line — Used for inequalities with < or > symbols, indicating points on the line are excluded from the solution.
Test point method — A technique using coordinates (often the origin) to determine which side of the boundary line to shade.
System of inequalities — Two or more inequalities considered simultaneously, where the solution region satisfies all conditions at once.
Vertices (corner points) — The intersection points where boundary lines meet, often required for optimization problems in linear programming contexts.
Core concepts
Understanding inequality symbols in two dimensions
When working with two variables, each inequality divides the Cartesian plane into two half-planes. The inequality symbol determines which half-plane represents the solution:
- y > mx + c means the region above the line y = mx + c
- y < mx + c means the region below the line y = mx + c
- x > a means the region to the right of the vertical line x = a
- x < a means the region to the left of the vertical line x = a
The symbols ≥ and ≤ include the boundary line itself (draw solid), while > and < exclude it (draw dashed).
Graphing a single linear inequality: Step-by-step method
Step 1: Draw the boundary line
Replace the inequality symbol with = and graph the resulting straight line. You need at least two points:
- Find the x-intercept (set y = 0)
- Find the y-intercept (set x = 0)
- Plot both points and draw the line
Determine line type: solid for ≤ or ≥, dashed for < or >.
Step 2: Identify the solution region using the test point method
Choose a test point not on the boundary line (the origin (0, 0) works best unless the line passes through it). Substitute the test point coordinates into the original inequality:
- If the inequality is TRUE, shade the region containing the test point
- If the inequality is FALSE, shade the region on the opposite side
Step 3: Label the region
Write the inequality near the shaded region or use the letter R to indicate the required region.
Graphing systems of linear inequalities
Many CXC CSEC questions require you to graph two or more inequalities simultaneously. The solution region is where all shaded areas overlap.
Method for systems:
- Graph each inequality separately using the step-by-step method above
- Use different shading patterns or directional arrows for each inequality (exam papers often specify this)
- Identify the region that satisfies ALL inequalities—this is typically a polygon
- Label vertices if required (find intersection points by solving pairs of equations simultaneously)
Common combinations tested:
- Two or three linear inequalities forming a triangular or quadrilateral region
- Inequalities combined with domain restrictions (x ≥ 0, y ≥ 0) to keep solutions in the first quadrant
- Real-world constraints modeled by inequalities
Converting word problems to inequalities
CXC examiners frequently present linear inequalities within practical contexts. Key phrases translate to mathematical symbols:
- "At least" → ≥
- "No more than" / "At most" → ≤
- "More than" / "Greater than" → >
- "Less than" / "Fewer than" → <
- "Maximum" → ≤
- "Minimum" → ≥
Example context: A farmer in St. Elizabeth, Jamaica plants x acres of sugarcane and y acres of pimento. If labour costs $500 per acre for sugarcane and $300 per acre for pimento, and the farmer has at most $6000 for labour, the inequality is:
500x + 300y ≤ 6000
Special cases and boundary conditions
Horizontal and vertical boundaries:
- y ≥ 3 represents the region on or above the horizontal line y = 3
- x < -2 represents the region to the left of the vertical line x = -2
Strict vs. non-strict inequalities:
Students often lose marks by drawing solid lines when dashed lines are required. Check the inequality symbol carefully before drawing.
Origin as a boundary point:
When the boundary line passes through (0, 0), select a different test point such as (1, 0), (0, 1), or (1, 1).
Reading solutions from graphs
CXC questions may provide a shaded region and ask you to:
- Write the inequality or inequalities that define it
- List integer coordinate pairs within the region
- Identify the maximum or minimum value of an expression within the region
Technique: Examine the boundary line's equation first (using two clear points), then determine the inequality symbol by checking a point within the shaded region.
Worked examples
Example 1: Graphing a single inequality
Question: On graph paper, using a scale of 2 cm to represent 2 units on both axes, draw the graph of the inequality 2x + y ≤ 6 for the domain 0 ≤ x ≤ 4. Shade the unwanted region.
Solution:
Step 1: Draw the boundary line 2x + y = 6
When x = 0: 2(0) + y = 6, so y = 6. Point: (0, 6) When y = 0: 2x + 0 = 6, so x = 3. Point: (3, 0)
Plot (0, 6) and (3, 0). Draw a solid line through these points (because the inequality includes ≤).
Step 2: Use the test point (0, 0)
Substitute into 2x + y ≤ 6: 2(0) + 0 ≤ 6 0 ≤ 6 ✓ TRUE
The origin satisfies the inequality, so the solution region contains (0, 0).
Step 3: Shade the unwanted region
Since the question asks to shade the unwanted region, shade the area above the line (away from the origin). The region below and including the line represents 2x + y ≤ 6.
Include the domain restriction by drawing vertical lines at x = 0 and x = 4, shading outside 0 ≤ x ≤ 4.
[2 marks for correct boundary line, 1 mark for appropriate line type, 2 marks for correct shading]
Example 2: System of inequalities with Caribbean context
Question: A vendor at the Port of Spain market sells coconuts and mangoes. She can carry at most 40 fruits. Coconuts cost $8 each and mangoes cost $5 each, and she wants to stock at least $200 worth of fruit. Let x represent coconuts and y represent mangoes.
(a) Write two inequalities, other than x ≥ 0 and y ≥ 0, to represent this information. (2 marks)
(b) On graph paper, using a scale of 2 cm to represent 5 units on both axes, show the region that satisfies all four inequalities. (6 marks)
Solution:
(a)
Total fruits: x + y ≤ 40
Total value: 8x + 5y ≥ 200
(b)
For x + y ≤ 40:
When x = 0: y = 40. Point: (0, 40) When y = 0: x = 40. Point: (40, 0)
Draw a solid line through (0, 40) and (40, 0).
Test (0, 0): 0 + 0 ≤ 40 ✓ TRUE Shade below the line (region containing origin).
For 8x + 5y ≥ 200:
When x = 0: 5y = 200, y = 40. Point: (0, 40) When y = 0: 8x = 200, x = 25. Point: (25, 0)
Draw a solid line through (0, 40) and (25, 0).
Test (0, 0): 8(0) + 5(0) ≥ 200, 0 ≥ 200 ✗ FALSE Shade above the line (region away from origin).
For x ≥ 0 and y ≥ 0: Restrict to the first quadrant.
Solution region: The quadrilateral bounded by the four inequalities in the first quadrant, with vertices approximately at (0, 40), (25, 0), (40, 0), and where the two lines intersect.
To find the intersection: Solve x + y = 40 and 8x + 5y = 200 simultaneously. From first equation: y = 40 - x Substitute: 8x + 5(40 - x) = 200 8x + 200 - 5x = 200 3x = 0, x = 0
The lines intersect at (0, 40). The feasible region is the quadrilateral with vertices (0, 40), (5, 32), (25, 15), and (25, 0).
[2 marks for inequalities, 3 marks for correct boundary lines with appropriate solid/dashed format, 3 marks for correctly identified solution region]
Example 3: Writing inequalities from a graph
Question: The shaded region R on a graph is bounded by the lines y = 2, x = 5, and y = x - 1. The region R is below y = x - 1, above y = 2, and to the left of x = 5. Write three inequalities that define region R. (3 marks)
Solution:
Below the line y = x - 1: y < x - 1 (or y ≤ x - 1 if the boundary is included)
Above the line y = 2: y > 2 (or y ≥ 2 if the boundary is included)
To the left of x = 5: x < 5 (or x ≤ 5 if the boundary is included)
[1 mark for each correct inequality]
Common mistakes and how to avoid them
Drawing the wrong line type — Students draw solid lines for strict inequalities (< or >) when dashed lines are required. Correction: Before drawing, circle the inequality symbol and note: < or > = dashed; ≤ or ≥ = solid.
Shading the wrong region — Forgetting to test a point or testing incorrectly leads to shading the opposite region. Correction: Always write out the test point substitution: "Test (0,0): [calculation] = TRUE/FALSE" then shade accordingly.
Misreading "shade the unwanted region" — Shading the solution region when the question asks for the unwanted region. Correction: Highlight command words in the question; if it says "unwanted," shade the area that does NOT satisfy the inequality.
Incorrect boundary line equations — Algebraic errors when rearranging inequalities like 3x - 2y < 6 into y = form. Correction: Rewrite as an equation first, then solve for y: 3x - 2y = 6 → -2y = -3x + 6 → y = (3/2)x - 3. Remember: dividing or multiplying by a negative number reverses the inequality symbol.
Not labeling axes or using wrong scales — Losing marks for poor graph presentation. Correction: Rule a title, label both axes clearly (x and y), mark scales as specified in the question (e.g., 2 cm to 2 units), and use a sharp pencil with a ruler.
Failing to consider all inequalities in a system — Identifying the wrong feasible region by overlooking one constraint. Correction: List all inequalities before graphing, tick each one as you draw it, and verify that your final region satisfies all conditions using a test point within it.
Exam technique for Relations, Functions and Graphs: Linear inequalities and regions in two variables
Command word "Draw" or "Show the region" — You must construct an accurate graph on provided graph paper using the specified scale. Marks are awarded for correct axes (1 mark), accurate boundary lines (1-2 marks each), appropriate line types (1 mark), and correct shading/region identification (1-2 marks). Allow 10-12 minutes for a full graphing question worth 6-8 marks.
"Write an inequality" — Extract the inequality from a worded constraint or graph. For word problems, identify the variables first, then translate phrases carefully. For graphs, determine the boundary equation using two clear points, then test a point in the shaded region to establish the inequality symbol. Typically worth 1-2 marks per inequality.
"Shade the unwanted region" — Many CXC questions ask for negative shading (the area that does NOT satisfy the inequality). Use light, neat pencil shading with a consistent pattern (parallel lines or cross-hatching). If you shade the wrong area, do not attempt to erase extensively; instead, outline the correct solution region clearly with a label "R."
Marks for accuracy — Graph questions award marks for precision: plot points accurately to within 2 mm of the correct position, use a ruler for all straight lines, and extend lines appropriately within the given domain. Carry a sharp pencil, ruler, and eraser to every exam.
Quick revision summary
Linear inequalities in two variables divide the Cartesian plane into regions. Graph the boundary line by replacing the inequality with =, using solid lines for ≤/≥ and dashed for </>;. Apply the test point method—substitute (0,0) into the original inequality to determine which side to shade. For systems, the feasible region satisfies all inequalities simultaneously. CXC exams test graphing accuracy, correct shading, and translating word problems to inequalities. Always label regions clearly and verify line types match the inequality symbols. Master the step-by-step method and practice with graph paper at the specified scales.