Optimization (Inequalities)

Optimization is the process of finding the best possible outcome — the maximum or minimum value of a quantity — given a set of constraints. In the context of linear inequalities, optimization means identifying which values of variables satisfy all given conditions (the feasible region) and then determining which point within that region gives the largest or smallest value of some target expression called the objective function. You encounter optimization problems constantly in real life: a farmer wants to maximize crop yield given limited land and water; a business wants to minimize costs while meeting production quotas; a student wants to maximize study efficiency within a fixed number of hours. These real-world situations translate naturally into systems of linear inequalities, and the techniques of linear programming — a mathematical method built directly on this idea — let us solve them systematically. At the high school level, optimization with linear inequalities focuses on two key ideas: graphing a system of inequalities to find the feasible region (the set of all points satisfying every constraint), and evaluating an objective function at the corner points (vertices) of that region. A fundamental theorem of linear programming guarantees that if an optimal solution exists, it always occurs at one of these corner points — so you never need to check infinitely many points.

Key takeaways
  • Define Variables and the Objective Function
  • Write the Constraints as Inequalities
  • Graph the Feasible Region
  • Identify the Corner Points (Vertices)
  • Evaluate the Objective Function at Each Vertex
  • State the Solution

Define Variables and the Objective Function

Identify what quantities you are controlling (the decision variables, typically x and y) and write the expression you want to maximize or minimize. This expression, called the objective function, is linear: P = ax + by for constants a and b. For example, if profit is 3perunitofproductAand3 per unit of product A and 5 per unit of product B, the objective function is P = 3x + 5y.

Write the Constraints as Inequalities

Translate every restriction into a linear inequality. Constraints often come from limited resources, minimum requirements, or physical limits. Always include non-negativity constraints x ≥ 0 and y ≥ 0 unless the problem states otherwise. For example: '2x + y ≤ 100 (labor hours), x + 3y ≤ 90 (material units), x ≥ 0, y ≥ 0.'

Graph the Feasible Region

Graph each inequality on the coordinate plane. For each constraint, graph the boundary line (as a solid line, since our inequalities use ≤ or ≥) and shade the appropriate half-plane. The feasible region is the intersection of all shaded areas — every point inside it satisfies all constraints simultaneously. If the feasible region is empty, the problem has no solution.

Identify the Corner Points (Vertices)

The corner points are where boundary lines of the feasible region intersect. Find them by solving pairs of boundary equations simultaneously. For example, if two boundaries are 2x + y = 100 and x + 3y = 90, solve this system: multiply the first equation by 3 to get 6x + 3y = 300, then subtract the second to get 5x = 210, so x = 42 and y = 16. Always include the intersections with the axes as additional corner points.

Evaluate the Objective Function at Each Vertex

Substitute the coordinates of each corner point into the objective function P = ax + by. Calculate the value at every vertex. The largest value is the maximum; the smallest is the minimum. The Corner Point Theorem (a consequence of the Fundamental Theorem of Linear Programming) guarantees that the optimal value of a linear objective function over a convex polygonal feasible region always occurs at a vertex. This is why you only need to check corner points — not every point in the region.

State the Solution

Report the values of x and y that give the optimal value, and state the optimal value itself. Always check that your solution actually lies inside the feasible region by verifying each constraint inequality. Include units in your answer where appropriate.

The Key Equation

P=c1x+c2ysubject toaix+biy≤di,  x≥0,  y≥0P = c_1 x + c_2 y \quad \text{subject to} \quad a_i x + b_i y \leq d_i, \; x \geq 0, \; y \geq 0
  • P — Objective Function Value (depends on context (dollars, units, etc.)): The quantity being maximized or minimized — profit, cost, area, or any linear combination of the variables.
  • c_1, c_2 — Objective Coefficients (same as P per unit of x or y): Constants that measure how much each variable contributes to the objective. For example, profit per unit produced.
  • x, y — Decision Variables (units of each product, resource, etc.): The quantities you control and optimize. Their values are what you solve for.
  • a_i, b_i — Constraint Coefficients (resource units per decision unit): Constants describing how much of each resource one unit of x or y consumes in constraint i.
  • d_i — Constraint Bound (total resource units available): The right-hand side limit of constraint i — the total amount of a resource available or required.

Worked Example

Problem. A school club makes two types of candles: pillar candles (x) and jar candles (y). Each pillar candle requires 2 hours of labor and earns 6profit.Eachjarcandlerequires3hoursoflaborandearns6 profit. Each jar candle requires 3 hours of labor and earns 8 profit. The club has at most 24 hours of labor available. They also need to make at least 2 pillar candles and at least 2 jar candles to meet pre-orders. How many of each type should they make to maximize profit?

x=9,  y=2,  Pmax⁡=$70x = 9,\; y = 2,\; P_{\max} = \$70

Answer. The club should make 9 pillar candles and 2 jar candles to earn a maximum profit of $70.

Figure
Diagram
Feasible Region for the Candle Problemxy123456789102466.7x = 2y = 22x+3y=24A(2,2)B(2, 20/3)C(9,2)FeasibleRegionP=70 (max)
Origins · 1 minGeorge Dantzig and how this was discovered
GD
George Dantzig
1914–2005 · American

George Dantzig invented the simplex method in 1947, a systematic algorithm for solving linear programming (optimization) problems with many variables and constraints — far beyond what can be graphed by hand. His work made it possible to solve enormous real-world optimization problems in logistics, economics, and engineering, forming the mathematical foundation that modern supply chains, airline scheduling, and resource allocation all depend on. His formulation of the standard linear programming problem — maximize c^T x subject to Ax ≤ b, x ≥ 0 — is the direct generalization of the two-variable systems studied in high school. As a graduate student at UC Berkeley, Dantzig arrived late to a statistics class and copied two problems from the board, assuming they were homework. He solved them and handed them in. His professor later told him those were actually famous unsolved problems in statistics — Dantzig had unknowingly solved them both!

  • Graphing Linear Inequalities — Optimization requires graphing a system of inequalities to find the feasible region; mastering single inequality graphs is the essential prerequisite.
  • Systems of Equations: Substitution Method — Finding corner points of the feasible region requires solving pairs of linear equations simultaneously, making systems of equations a direct building block.
  • Compound Inequalities — Constraint systems in optimization are essentially compound inequalities that must all be satisfied at once — understanding compound inequalities deepens comprehension of the feasible region.
  • Slope-Intercept Form of a Line — Every constraint boundary and the objective function itself is a linear equation; fluency with slope-intercept form is needed to graph boundaries accurately.