Graphing Constraint Lines and Feasible Regions

S4 Mathematics — Term 1, Topic 3

In this lesson:

• Draw boundary lines using the intercept method
• Shade the correct half-plane using the test-point method
• Identify and label the feasible region F

What You Will Learn Today

By the end of this lesson:

1. Draw boundary lines for linear inequalities using the intercept method
2. Shade the correct half-plane using the test-point method
3. Identify and label the feasible region of a system of inequalities

Which Carpenter Plans Actually Work?

= tables, = chairs, four constraints from Lesson 2.

• satisfies all four — so does , so does
• There are infinitely many valid plans; you cannot check them all by hand
• How do you see every valid plan at once?

Five Plans Against One Constraint

The Boundary Line: Where Equality Holds

Every inequality has a boundary — the line separating valid from invalid:

• Replace with to get the boundary equation
• → boundary:
• One side: boards used ✓ — other side: ✗

Drawing the First Boundary Line

• Set : → ; set : →
• Connect and label

Graph the Second Boundary Line

• Set : ___; set : ___
• Plot both intercepts, connect, and label on the same axes

Both Boundary Lines Drawn, No Shading

• Both lines drawn and labelled
• Intercepts: , , , — all four marked
• Draw both lines before shading either half-plane

Quick Check: Find the Intercepts

Find the intercepts of .

• Set and solve: ___
• Set and solve: ___

Name the coefficient you divide by at each step before you divide.

A Line Splits the Plane in Two

Every boundary line divides the plane into two half-planes:

• One side: boards used — the plan fits ✓
• Other side: boards used — the plan fails ✗

To find which side satisfies the inequality — test one point.

Testing a Point: First Constraint

Test point: . Substitute: . Is ? Yes — tick at .

Shade the side containing . Draw the boundary solid ( includes the line).

Test the Second Constraint the Same Way

Test point: . Substitute: . Is ? Yes — tick at .

Shade the origin side with a solid boundary. The same three-step procedure works for every constraint.

Which Shading Is Correct? Predict First.

After testing in and finding ✓:

• Student A shades the region containing the origin
• Student B shades the region not containing the origin

Which student is correct, and what rule tells you so?

Commit to an answer before the next slide.

When the Boundary Passes Through the Origin

If the boundary passes through , substituting gives — a tie, not a verdict.

Choose as the test point instead.

Example: boundary passes through the origin.
Test : . Is ? No ✗ → shade the other side.

Solid or Dashed? The Boundary Rule

• : solid boundary — gives ✓ (boundary included)
• : dashed boundary — gives ✗ (boundary excluded)
• Rule: or → solid; or → dashed

Quick Check: Boundary Through the Origin

The constraint is .

Its boundary passes through the origin.

1. Which test point should you use instead of ?
2. Substitute and determine which half-plane to shade.

Write your answer before advancing.

Two Half-Planes — Where Do Both Hold?

The diagram now shows two shaded half-planes for the carpenter system.

A valid plan — one the carpenter can actually make — must satisfy both constraints at the same time: AND .

Where on this diagram does that hold?

The Feasible Region: Intersection, Not Union

The feasible region is the set of all points satisfying every constraint simultaneously:

• Graphically: where all shaded half-planes overlap (the intersection)
• Not the union — satisfying one constraint is not enough
• NCDC convention: label the overlap F; cross-hatch the excluded areas outside

The Carpenter's Feasible Region, Labelled F

Region F: every point satisfies , , , and simultaneously. Points outside at least one constraint are cross-hatched.

Union or Intersection? Make Your Prediction.

The region covering everything shaded by at least one constraint is the union of the four half-planes.

• Does this region equal F?
• If not, pick a point inside the union but outside F, and substitute it into all four constraints. What do you find?

Test Two Points: Inside F or Not?

For each point, substitute into all four constraints — use algebra, not the graph.

Point A: — does it satisfy ? Try it.

Point B: — check all four constraints.

Write out each substitution before advancing.

Is (3, 2) Inside Region F?

Substitute into all four constraints:

1. — Is it ?
2. — Is it ?
3. Is ? Is ?

What does your answer tell you about where (3, 2) sits on the diagram?

Your Turn: Build the Full Graph

On blank axes:

1. Find intercepts and draw both boundary lines
2. Test-point method — shade each half-plane
3. Label F, cross-hatch outside

No diagram provided — start from scratch.

Four Mistakes to Watch For

M1: After , divide by 's coefficient — not the zeroed one.

M2: Tick → shade that side; cross → shade the other.

M3: F is the overlap — not the union.

M4: or → solid; or → dashed.

What the Feasible Region Now Gives You

✓ A system of inequalities carves out region F — the set of valid plans.

✓ Every point in F satisfies all constraints simultaneously.

✓ Corner points already visible: , , , .

Lesson 4: which corner maximises the carpenter's profit?