Lesson 5: Optimization

Property

To solve applications with linear inequalities, first identify what you are looking for and assign a variable. Translate the problem into an inequality by representing the situation. Solve the inequality, and ensure the answer makes sense in the context of the problem, rounding if necessary.

Examples

• A grant is 3,000 dollars for science kits. Each kit costs 75 dollars. What is the maximum number of kits, $k$, you can buy? The inequality is $75k \leq 3000$. Solving gives $k \leq 40$. You can buy a maximum of 40 kits.
• Your phone bill is 25 dollars plus 0.15 dollars per text, $t$. To keep your bill no more than 40 dollars, the inequality is $25 + 0.15t \leq 40$. Solving gives $t \leq 100$. You can send at most 100 texts.
• To earn a profit of at least 500 dollars, revenue must exceed costs by that amount. If you sell items for 10 dollars each and expenses are 400 dollars, how many items, $b$, must you sell? The inequality is $10b - 400 \geq 500$. Solving gives $b \geq 90$ items.

Explanation

Use inequalities to solve real-world problems about budgets, profits, or limits. Set up an expression for the total cost or amount and compare it to the limit using an inequality sign. Then, solve for your variable.

Property

To find the vertices (corner points) of a feasible region for optimization:

1. Graph each constraint inequality to form the feasible region boundary
2. Identify where boundary lines intersect by solving pairs of equations
3. Check that each intersection point satisfies all constraints
4. The vertices are the corner points where the feasible region changes direction

These vertices are critical because the optimal solution to a linear programming problem always occurs at a vertex of the feasible region.

Examples

Property

For a linear programming problem with a bounded feasible region, the optimal value (maximum or minimum) of a linear objective function $f(x,y) = ax + by$ occurs at one of the vertices (corner points) of the feasible region.

Examples