3(50 - 2y) + 2y \leq 90 \\
Understanding the Inequality: 3(50 - 2y) + 2y ≤ 90 A Comprehensive Breakdown for Beginners with Solving Strategies
Introduction
Mathematical inequalities play a crucial role in solving real-life problems, optimizing resources, and modeling constraints—especially in algebra. One common type is linear inequality, such as:
3(50 - 2y) + 2y ≤ 90
At first glance, this inequality may seem complex, but with a step-by-step approach, anyone can simplify, solve, and interpret it. Whether you're studying math in school, preparing for standardized tests, or applying algebra to practical scenarios, understanding how to work with such expressions is invaluable. In this article, we’ll break down the inequality, solve it systematically, and explore its real-world implications.
Step 1: Simplify the Expression
We begin by expanding and combining like terms in the expression 3(50 - 2y) + 2y ≤ 90.
Expand the parentheses: 3 × 50 = 150 3 × (-2y) = -6y So, 3(50 - 2y) = 150 - 6y
Now substitute back: 150 - 6y + 2y ≤ 90
Combine like terms: -6y + 2y = -4y
So the inequality becomes: 150 - 4y ≤ 90
Step 2: Isolate the Variable Term
Next, subtract 150 from both sides to isolate the term with y: 150 - 4y - 150 ≤ 90 - 150 Simplify: -4y ≤ -60
Step 3: Solve for y
Now divide both sides by -4. Remember: dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. y ≥ (-60) ÷ (-4) y ≥ 15
Final Solution
The solution to the inequality 3(50 - 2y) + 2y ≤ 90 is: y ≥ 15
What Does This Mean?
This result tells us that y must be 15 or greater to satisfy the original inequality. In practical terms, if y represents a measurable quantity—such as a daily limit, production unit, or cost factor—this constraint ensures values stay within acceptable bounds.
For example:
- If y is the number of items produced per day, production must be ≥15 units daily to meet capacity or regulatory limits.
- In budgeting, if y stands for spending beyond a base 50 minus cost terms, then spending 15 or more results in a balanced or acceptable financial outcome under the model.
Visual Interpretation: Number Line Representation
In inequality graphs, y ≥ 15 is shown as a line crossing 15 with a closed circle (indicating inclusion) and shading to the right, representing all values greater than or equal to 15.
Why This Inequality Matters
Understanding and solving such inequalities enables clearer decision-making under constraints. They appear in fields like:
- Operations Research: Optimizing resource allocation
- Economics: Modeling production costs and revenue
- Engineering: Ensuring design parameters stay within safe limits
- Finance: Managing debt-to-income ratios
Practice Problem
Try solving this variation: 3(40 - 2y) + y ≥ 70 Use the same methods—expand, combine, isolate, and apply the sign rule for negatives—to find the new solution interval for y.
Conclusion
The inequality 3(50 - 2y) + 2y ≤ 90 simplifies beautifully to y ≥ 15, offering a clear constraint with meaningful real-world relevance. Whether you're a student mastering algebra or a professional applying mathematical models, mastering inequalities like this strengthens your analytical toolkit.
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