Understanding the Context

What is the Inequality 2x + 3y ≤ 120?

The expression 2x + 3y ≤ 120 is a linear inequality in two variables, commonly encountered in operations research, optimization, and linear programming. Here, x and y represent variables (often quantities, costs, times, or resources), and the inequality expresses a constraint: the combined weighted usage of x and y must not exceed 120 units.

Interpretation:

• x and y are non-negative variables (x ≥ 0, y ≥ 0).
  
• The expression models limitations such as budget boundaries, time constraints, material availability, or capacity limits in manufacturing, scheduling, or budgeting.

Key Insights

Solving the Inequality

To work with 2x + 3y ≤ 120 effectively, it’s useful to understand how to manipulate and visualize it:

Step 1: Graphical Representation

Plot the line 2x + 3y = 120 in the coordinate plane:

• When x = 0, y = 40
  
• When y = 0, x = 60
  These two intercepts define a straight line, and the inequality describes a shaded region below and including this line in the first quadrant (since x, y ≥ 0).

Step 2: Finding Feasible Solutions

The solution set includes all (x, y) pairs such that the point lies:

• On or below the line 2x + 3y = 120
  
• And in the first quadrant x ≥ 0, y ≥ 0

This feasible region is a triangle with vertices at (0,0), (60,0), and (0,40). Resources or networks modeled by such inequalities lie within this bounded region.

Final Thoughts

Real-World Applications

1. Resource Allocation

Suppose x represents units of Product A and y units of Product B, each requiring 2 hours and 3 hours of labor, respectively, with only 120 hours available. This inequality ensures total labor does not exceed capacity.

2. Budget Constraints

If x = marketing spend and y = operational cost, the inequality limits total expenditure to 120 units.

3. Production Planning

Manufacturers use such models to determine combinations of products that maximize output under material or machine limits.