Résoudre \( 2x + 4y \leq 100 \) :

["# Optimize Your Resource Allocation: Solving the Inequality ( 2x + 4y \leq 100 )", "When managing limited resources—whether in budgeting, production planning, or time management—inequalities like ( 2x + 4y \leq 100 ) become crucial tools for decision-making. This fundamental linear inequality helps businesses, educators, and individuals optimize efficiency without exceeding constraints. In this article, we explore how to solve ( 2x + 4y \leq 100 ), interpret its meaning, and apply it in practical scenarios to make smarter choices.", "---", "## Understanding the Inequality ( 2x + 4y \leq 100 )", "The inequality ( 2x + 4y \leq 100 ) defines a region in the coordinate plane where the sum of weighted variables ( x ) and ( y ), scaled by coefficients 2 and 4 respectively, does not exceed 100. Interpreted in real-world terms, ( x ) and ( y ) often represent quantities like units produced, hours worked, or costs—subject to budget, material, or time limits.", "### Rewriting the Inequality for Clarity", "To better analyze the constraint:", "[\n2x + 4y \leq 100\n]", "Divide every term by 2 to simplify:", "[\nx + 2y \leq 50\n]", "This simplified form clarifies that the combined impact of ( x ) and ( y ) is bounded: adding twice as much ( y ) as ( x ) still consumed no more than 50 units of capacity.", "---", "## Graphing the Feasible Region", "To solve ( x + 2y \leq 50 ), we first consider its boundary line:", "[\nx + 2y = 50\n]", "### Step 1: Find intercepts\n- x-intercept: Set ( y = 0 \Rightarrow x = 50 )\n- y-intercept: Set ( x = 0 \Rightarrow y = 25 )", "### Step 2: Sketch the line and region\nDraw a straight line connecting ( (50, 0) ) and ( (0, 25) ). Since the inequality includes equality (( \leq )), the line is solid. The region ( x + 2y \leq 50 ) lies below and including this line.", "Shading the solution region means selecting the area containing all points where ( x + 2y \leq 50 ), such as the origin ( (0,0) ), which satisfies the inequality.", "---", "## Applications and Real-World Examples", "### 1. Budget and Cost Management\nSuppose ( x ) is the number of units produced at cost ( $2 ), and ( y ) represents marketing spend at $4 per unit. The total must stay under $100. The inequality ensures you remain within budget while optimizing output.", "### 2. Production Planning\nFactories limit resource use—like raw materials or labor hours—via constraints. Managing ( x ) and ( y ) under ( x + 2y \leq 50 ) helps maximize production volume or product lines while respecting constraints.", "### 3. Time Allocation\nIf ( x ) represents hours spent on project A and ( y ) on project B, total work time mustn’t exceed 50 hours. This helps professionals balance workloads and avoid burnout.", "---", "## Solving for Optimal Values: Objective Functions", "In optimization, ( x ) and ( y ) often represent decision variables in a function to be maximized or minimized, subject to ( 2x + 4y \leq 100 ).", "For example, suppose your goal is to maximize total output ( Z = 3x + 5y ). Solving this as a linear programming problem yields the optimal combination of ( x ) and ( y ) within constraints.", "Using the Simplex Method or Graphical Analysis, you evaluate corner points of the feasible region—such as ( (0,0) ), ( (50,0) ), ( (0,25) ), and intersection with other constraints—to find the maximum ( Z ).", "---", "## Tips for Effective Problem Solving", "- Plotting helps visualize constraints—use graph paper or graphing software to sketch lines and shaded regions.\n- Test corner points to find optimal solutions, especially when objectives involve maximization or minimization.\n- Combine with other constraints when modeling complex systems; real-life problems often involve multiple limitations.\n- Use technology: Solvers and spreadsheet tools can automate computation and allow scenario testing.", "---", "## Summary", "The inequality ( 2x + 4y \leq 100 ) (or ( x + 2y \leq 50 )) is a powerful linear model for resource management. Understanding its solution region enables smarter, data-driven decisions in budgeting, production, scheduling, and beyond. By simplifying, graphing, and applying objective functions, you transform abstract constraints into actionable strategies—solving not just equations, but real challenges.", "---", "Keywords: résoudre ( 2x + 4y \leq 100 ), inequality solutions, linear programming, resource allocation, graph linear inequality, budget constraint, production optimization, objective function, real-world constraint modeling.", "---", "Whether you're a student, manager, or planner, mastering linear inequalities like this equips you to optimize outcomes practically and effectively."]