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Learn Index · Patterns and Algebra Grade 11–12

Systems of inequalities

Graphing two or more inequalities; the answer is where the shading overlaps.

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Graphing one linear inequality

An inequality like y > 2x − 1 describes a half-plane: all the points on one side of a line.

  1. Graph the boundary line y = 2x − 1.
    • Use a dashed line for < or > (points on the line are not included).
    • Use a solid line for ≤ or ≥ (they are included).
  2. Pick a test point not on the line, often (0, 0).
  3. If it makes the inequality true, shade its side; otherwise shade the other side.

Test (0, 0): 0 > 2(0) − 1 → 0 > −1 ✓. Shade the side containing the origin.

Systems of inequalities

A system of inequalities has two or more conditions. The solution is the feasible region: where all the shaded regions overlap.

Example:

  • y ≤ −x + 6
  • y > x − 2
  • x ≥ 0
  1. Graph y = −x + 6 (solid); shade below.
  2. Graph y = x − 2 (dashed); shade above.
  3. Graph x = 0 (solid); shade to the right.
  4. The overlapping region is the solution.

Check a point in the region, like (1, 2):

  • 2 ≤ 5 ✓
  • 2 > −1 ✓
  • 1 ≥ 0 ✓

Vertices of the region

The corners (vertices) of the feasible region are where boundary lines intersect. Find them by solving pairs of equations.

From the example: y = −x + 6 and y = x − 2 meet where −x + 6 = x − 2 → x = 4, y = 2. Vertex: (4, 2).

Real-world constraints

Example: A bakery makes muffins (x) and cookies (y).

  • Oven space: x + y ≤ 100
  • At least 20 muffins: x ≥ 20
  • Cookies at least twice the muffins: y ≥ 2x
  • No negatives: y ≥ 0

The feasible region shows every possible production plan.

Linear programming

Linear programming finds the best value of a quantity (like profit) inside a feasible region. The best value always occurs at a vertex.

Example: Profit P = 2x + 3y with vertices (0, 0), (0, 40), (30, 20), (40, 0).

Vertex P
(0, 0) 0
(0, 40) 120
(30, 20) 120
(40, 0) 80

Maximum profit: 120, at (0, 40) or (30, 20) (and every point on the edge between them).

Worked examples

Example 1: Is (2, 3) in the region y ≥ x and y < 5? 3 ≥ 2 ✓, 3 < 5 ✓ → yes.

Example 2: Dashed or solid for x + y < 8? Dashed.

Example 3: Where do x + y = 10 and y = 4 meet? (6, 4).

Common mistakes

  • Solid vs dashed lines mixed up.
  • Shading the wrong side: always use a test point.
  • Using the origin as a test point when it’s on the line.
  • Forgetting non-negativity constraints in real problems.

Try it yourself

  1. Which side of y > 3x + 1 does (0, 0) lie on: shaded or not?
  2. Find the vertex where y = 2x and x + y = 9 meet.
  3. Is (1, 1) a solution of x + 2y ≤ 4 and x − y > 0?
  4. Maximize P = 5x + 4y at vertices (0, 0), (0, 6), (4, 3), (5, 0).

Answers: 1) not shaded (0 > 1 is false) 2) (3, 6) 3) No: 1 − 1 = 0 is not > 0. 4) 32 at (4, 3)

Quick recap

  • Graph each boundary line (solid or dashed) and shade the correct half-plane.
  • The feasible region is where all shadings overlap.
  • Linear programming: the optimum is at a vertex.