Inequalities & Optimization

1 Solving Linear Inequalities

Solving inequalities is very similar to solving equations, with one critical difference:

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The Golden Rule: When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

One-Variable Inequalities

Isolate the variable using the same steps as equations.
Check: did you divide/multiply by a negative? If yes, flip the sign.
Write your answer in interval notation or on a number line.

Simple example

3x + 4 < 22 → 3x < 18 → x < 6

With negative division

−5x ≥ 5 → x ≤ −1 (sign flips because ÷ −5)

Double inequality

13 ≤ 2x − 5 ≤ 21 → 18 ≤ 2x ≤ 26 → 9 ≤ x ≤ 13

2 Graphing Linear Inequalities

One Variable (on number line)

Closed dot (●) for ≤ or ≥ (endpoint included).
Open dot (○) for < or > (endpoint excluded).
Shade in the direction that satisfies the inequality.

Two Variables (on Cartesian plane)

Step 1: Graph the boundary line y = mx + b.
Step 2: Use a solid line for ≤ or ≥ (included), a dashed line for < or > (excluded).
Step 3: Use a test point (usually origin (0, 0)) — substitute into the inequality.
Step 4: If the test point satisfies the inequality, shade the side containing it. Otherwise shade the other side.

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Example: Graph y ≥ 2x + 1.
Draw solid line through (0,1) and (2,5).
Test (0,0): 0 ≥ 1 is false → shade AWAY from origin (the upper region).

3 Systems of Linear Equations (by graphing)

A system is two or more equations with the same unknowns. The solution is the ordered pair (x, y) that satisfies ALL equations simultaneously.

Graph each line using a table of values.
Find the intersection point — that is the solution.
Note: if lines are parallel, no solution; if same line, infinite solutions.

4 Systems of Inequalities

Graph each inequality separately. The solution is the region where all shaded areas overlap.

Graph each boundary line (solid or dashed).
Shade each inequality separately.
The overlapping region is the solution set — called the polygon (or region) of constraints.
The vertices (corner points) of this polygon are key for optimization.

5 Linear Programming (Optimization)

Linear programming finds the maximum or minimum value of an objective function, subject to constraints (a system of inequalities).

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Fundamental theorem: The optimal value (max or min) of a linear objective function over a polygon of constraints always occurs at a vertex of the polygon.

Steps to Solve an Optimization Problem

Step 1: Define variables (x = ?, y = ?).
Step 2: Write the objective function (e.g. Revenue = 20x + 30y).
Step 3: Write the constraint inequalities.
Step 4: Graph the constraints → find the polygon of constraints.
Step 5: Find the coordinates of each vertex (solve the boundary equations as a system).
Step 6: Substitute each vertex into the objective function.
Step 7: The largest (or smallest) value is the optimum. State the answer in context.

Multiple Optimal Solutions

If two adjacent vertices give the same objective value, then every point on the edge between them is also optimal.

✏️

Example: Salon earns $20/haircut (x) and $30/colouring (y). Objective: Max = 20x + 30y.
Vertex A(3,2): $120 · B(3,8): $300 · C(8,3): $250
Maximum is $300 at B(3,8) → do 3 haircuts + 8 colourings.

Open vs Closed Vertices

If a vertex touches a dashed boundary line, it is NOT included in the solution set — it cannot be an optimal solution.

6 Common Mistakes

Mistake	Fix
Forgetting to flip inequality when dividing by negative	Always check: did you divide or multiply by a negative? If yes, reverse the sign.
Wrong line type when graphing	< or > → dashed (excluded). ≤ or ≥ → solid (included).
Wrong shading direction	Use a test point! Substitute (0,0) and check which side satisfies the inequality.
Not finding ALL vertices of the polygon	Every intersection of two boundary lines is a potential vertex. Find them all.
Checking only some vertices	You MUST evaluate the objective function at ALL vertices — the maximum or minimum could be anywhere.