Represent Constraints with Systems

Lesson 3 of 3: Creating Equations

In this lesson:

• Write systems of inequalities from context
• Graph feasible regions using the test-point method
• Identify viable and non-viable solutions

What You Will Be Able to Do

1. Write each constraint as an equation or inequality
2. Graph a system and shade the feasible region
3. Test whether a point satisfies all constraints simultaneously
4. Explain why a point outside the region is non-viable
5. Read boundary lines as solid or dashed

Recall: Inequalities and Two-Variable Equations

• "At most" → ; "At least" →
• Two-variable equation: all pairs satisfying it form a line
• Test a point: substitute into an inequality to check if it holds

Can you translate "no more than " into an inequality?

Meal Planning: Two Foods, Two Requirements

You're planning a meal for 10 people on a $30 budget. The meal needs at least 40g of protein. Two options:

• Food A: 10g protein per lb, $2/lb
• Food B: 5g protein per lb, $4/lb

How do you find a combination that satisfies both requirements at once?

Which Combinations Pass Both Conditions?

Constraints: Every Condition Must Hold

• A constraint is a condition every valid solution must satisfy
• Real problems have multiple constraints — all must hold at the same time
• The set of all solutions satisfying every constraint: the feasible region

Writing the Full System: Food Example

= lb of Food A, = lb of Food B

1. Protein:
2. Budget:

Why You Must Include Non-Negativity

• Real quantities can't be negative: include and
• These restrict the feasible region to the first quadrant
• Without them, looks valid — but you can't buy negative pounds of food

List Every Constraint Before Writing Algebra

Product A (2 hrs), Product B (3 hrs). At most 120 hrs, at least 15 units total, at most 40 of B.

• Time:
• Units:
• Cap:
• Non-neg:

Each Inequality Defines a Half-Plane

• A linear inequality in two variables divides the plane into two half-planes
• The boundary line is where equality holds:
• Points on one side satisfy the inequality; points on the other side don't

Intersection Not Union: All Must Hold

Feasible = the overlap where every half-plane's shading agrees.

Three Steps to Shade One Constraint

Constraint:

Step 1: Graph — solid line (≥ includes boundary)

Step 2: Test : ? No

Step 3: Origin fails → shade away from the origin (above the line)

Two Constraints, One Feasible Region

Region F satisfies both and simultaneously.

Solid or Dashed: The Symbol Decides

• ≤ or ≥ → solid line — the boundary is included; spending exactly $30 satisfies "at most $30"
• < or > → dashed line — the boundary is excluded; exactly 5 fails "strictly more than 5"

Solid or Dashed? Which Side?

For each inequality below, state: (1) solid or dashed boundary, and (2) which direction to shade.

• A:
• B:

Test the origin for each. Decide before advancing.

Answers: A: solid, shade below; B: dashed, shade above

All Four Constraints: First-Quadrant Region

• Protein: — shade above
• Budget: — shade below
• , — first quadrant only

Every point inside the bounded region is a valid meal combination.

Mathematical vs. Contextual Viability Differ

Only whole units can be manufactured. Is valid?

• Mathematical: satisfies all inequalities — inside the feasible region ✓
• Contextual: can't manufacture 12.7 units ✗

A point can pass all the algebra and still fail the real-world check.

Testing Which Manufacturing Points Are Viable

Two Kinds of Invalid Solutions

Always check both: algebra first, then real-world sense.

Protein Rises to 60g: Larger or Smaller Region?

Does the feasible region get larger or smaller? Which direction does it shift?

Answer: Smaller — the protein constraint tightens, cutting away combinations that used to pass. The boundary line shifts outward, reducing the overlap area.

Corner Points: Solving Pairs of Boundary Lines

• Protein ∩ budget: solve and
• Protein ∩ -axis: ,
• Budget ∩ -axis: ,

The Feasible Region Tells the Full Story

• Every point inside the feasible region is a valid plan — it satisfies all constraints simultaneously
• Every point outside has broken at least one rule
• The constraints drew the boundary; you choose the plan

The feasible region is the picture of everything that works at once.

Which Corner Has the Lowest Meal Cost?

Find the Error: Which Half-Plane Is Wrong?

A student shaded below the protein line for .

What mistake did the student make?

Answer: The protein constraint uses ≥. Test : ? No — origin fails, so shade away from origin. The student shaded toward the origin — the wrong side.

Four Errors to Watch for Every Time

Wrong shading — origin fails? Shade away from it

Union instead of intersection — shade all constraints, then find the overlap

Missing non-negativity — always include and

Requiring integers in continuous contexts — time and weight allow non-integer solutions

Everything That Works at Once: Next Steps

The feasible region is the picture of everything that works at once.

• Every point inside satisfies all constraints simultaneously
• Every point outside breaks at least one rule

Coming next: Which plan is best? Linear programming finds the corner that maximizes profit or minimizes cost.