Matériel : \( 3x + 2y \leq 90 \)

["# Optimize Operations with the Linear Inequality ( 3x + 2y \leq 90 )", "The inequality ( 3x + 2y \leq 90 ) is a powerful mathematical model widely used in operations research, resource planning, and optimization problems—particularly in business, logistics, and engineering. Understanding and effectively applying this constraint helps decision-makers maximize efficiency, manage resources, and solve real-world logistical challenges.", "## What Is ( 3x + 2y \leq 90 )?", "At its core, the inequality ( 3x + 2y \leq 90 ) represents a linear constraint limiting the combined usage of two variables ( x ) and ( y ) subject to a total resource cap—in this case, 90. Depending on context, these variables often symbolize time, materials, budget allocations, or scaled workloads:", "- ( x ) and ( y ) might denote hours spent on two different tasks, quantities of two ingredients, or dollars allocated to two projects.\n- The coefficients (3 and 2) reflect the resource “cost” per unit of each variable, indicating how much each contributes to the total.\n- The constant 90 sets the upper boundary, ensuring total consumption never exceeds available capacity.", "## Applications in Business and Operations", "This inequality models scenarios where resource limits impact strategic choices. Here are common sectors where ( 3x + 2y \leq 90 ) plays a vital role:", "### 1. Production Planning", "Manufacturers use this constraint to determine feasible production combinations. For example:", "- ( x ): number of units of Product A\n- ( y ): number of units of Product B\n- Resource: total machine hours ≤ 90 per shift\n- Cost factor: Product A requires 3 hours, Product B requires 2 hours", "This ensures production stays within operational limits, avoiding overloading machinery.", "### 2. Budget Allocation", "Organizations must balance spending across two departments or projects:", "- ( x ): marketing budget\n- ( y ): R&D budget\n- Geographic or operational limit sets maximum combined spend at 90 units (e.g., $90,000).", "The inequality helps avoid exceeding financial caps while optimizing return.", "### 3. Supply Chain and Logistics", "In route planning or warehouse storage, ( 3x + 2y \leq 90 ) may represent combined weight, volume, or time constraints:", "- ( x ): weight of cargo A\n- ( y ): volume of cargo B\n- Carrier limits: total weight capacity or time available", "This prevents under- or overloading transport resources.", "## Solving the Inequality: Feasibility and Optimization", "The inequality defines a feasible region on a two-dimensional plane where other practical conditions (e.g., ( x \geq 0, y \geq 0 )) apply. Visualization shows a triangular or polygonal area bounded by:", "- The intercepts: when ( x = 0 \Rightarrow y \leq 45 ); when ( y = 0 \Rightarrow x \leq 30 ).\n- The line itself: ( 3x + 2y = 90 ), connecting (30, 0) and (0, 45).", "### Graphical Analysis", "- Plot intercepts to outline the feasible zone.\n- Shade the region ( 3x + 2y \leq 90 ), including boundary (≤ symbol).\n- Useful for decision-makers to instantly identify allowable combinations.", "### Optimization Using Linear Programming", "In operations research, the inequality is often part of a linear programming (LP) problem aiming to maximize or minimize a function—such as profit or cost:", "[\n\ ext{Maximize } Z = px + qy \quad \ ext{subject to} \quad 3x + 2y \leq 90, \ x \geq 0, \ y \geq 0\n]", "Where ( p ) and ( q ) reflect profitability or efficiency per unit. The constraint ( 3x + 2y \leq 90 ) limits feasible solutions, focusing optimization on practical outcomes.", "## Tips for Practical Use", "- Calibrate coefficients (3 and 2) with real data for accurate modeling.\n- Combine with other constraints (e.g., minimum/maximum thresholds) to specify exact operational requirements.\n- Use computational tools like Excel Solver, Python’s PuLP, or specialized LP solvers to automate analysis.\n- Visualize solutions—graphs aid interpretation and communication in teams.", "## Conclusion", "The inequality ( 3x + 2y \leq 90 ) is far more than an abstract expression—it is a cornerstone in optimization that guides efficient resource use across industries. By accurately representing limits and enabling systematic evaluation of possibilities, it empowers professionals to make data-driven, impactful decisions under constraints. Whether designing production schedules, planning budgets, or streamlining logistics, mastering this model translates directly into smarter, more effective operations.", "---", "### Keywords:\nlinear programming, optimization inequality, 3x + 2y ≤ 90, resource constraint modeling, operational research, business optimization, industrial logistics, production scheduling, linear constraint, mathematical modeling", "Enable better decision-making. Master ( 3x + 2y \leq 90 ) today."]