Linear Inequalities & Programming

Linear programming uses inequalities to describe limits, then finds the best possible outcome inside those limits. The shaded region is not decoration; it represents all values that satisfy every condition at the same time.

For CSEC, these problems usually involve resources such as time, money, materials, or production limits. Define the variables, write the inequalities from the wording, shade the feasible region, and test corner points when maximising or minimising. Each step explains the model.

Inequalities in One Variable¶

An inequality shows a relationship where something is greater than, less than, greater than or equal to, or less than or equal to something else.

Symbols:

• $>$ : greater than
• $<$ : less than
• $\geq$ : greater than or equal to
• $\leq$ : less than or equal to

Solving One-Variable Linear Inequalities¶

Solve like equations, BUT flip the inequality sign when multiplying or dividing by a negative number.

Inequalities in Two Variables¶

An inequality like $y > 2x + 1$ represents a region (not just a line).

Graphing Two-Variable Inequalities¶

Step 1: Graph the boundary line $y = 2x + 1$

• Use dashed line if inequality is $<$ or $>$ (not included)
• Use solid line if inequality is $\leq$ or $\geq$ (included)

Step 2: Shade the appropriate region

• For $y > 2x + 1$: shade ABOVE the line
• For $y < 2x + 1$: shade BELOW the line
• Test a point if unsure: pick $(0, 0)$ and check if it satisfies the inequality

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What Is Linear Programming?¶

Linear programming is a method to find the best solution (maximum or minimum) when you have:

• An objective function (what you want to maximize or minimize)
• Constraints (limitations or restrictions, usually linear inequalities)

Real-world applications:

• Maximize profit subject to resource limits
• Minimize cost subject to production requirements
• Optimize resource allocation

The Feasible Region¶

The feasible region is the area that satisfies all constraints simultaneously.

To find it:

1. Graph each constraint as a linear inequality
2. Shade the region that satisfies ALL constraints
3. The overlapping region is the feasible region

Optimization — Finding the Best Solution¶

The optimal solution occurs at a vertex (corner point) of the feasible region.

Method:

1. Find all corner points of the feasible region
2. Evaluate the objective function at each corner point
3. Choose the point that gives the maximum or minimum value