Optimize Your Linearity: Understanding and Solving the Inequality 2x + 3y ≤ 120

In mathematics, inequalities like 2x + 3y ≤ 120 serve as foundational tools across disciplines, from business planning to logistics and resource allocation. This article explores the significance of this linear inequality, how to solve and interpret it, and how it can help maximize efficiency in various real-world applications.

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.
2x + 3y \leq 120

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.

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All are non-negative. The sum of roots is $ 0 + 4 + 9 = 13 $.
Thus, the sum of all real solutions is $ \boxed{13} $.
Question: A museum curator uses a digital model where the position $ s(t) $ of a rotating exhibit arm is given by $ s(t) = t - \frac{t^5}{5} $. Define $ c_n $ by $ c_1 = \frac{1}{2} $ and $ c_{n+1} = s(c_n) $. Compute $ c_3 $.

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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.

Maximizing Value Under Constraints

In advanced scenarios, the goal shifts from merely satisfying the inequality to optimizing an objective—like profit or production—subject to 2x + 3y ≤ 120. This transforms the problem into a linear programming (LP) model:

Maximize:
 P = c₁x + c₂y (e.g., profit or utility)
Subject to:
 2x + 3y ≤ 120
 x ≥ 0, y ≥ 0

Using graphical or algebraic methods (like the Simplex algorithm), one identifies corner-point solutions to determine the optimal (x, y) pair that maximizes P.