Understanding the Context

The expression 6y + 2y ≤ 90 represents a simplified linear constraint where:

• y is the variable representing a measurable quantity (e.g., hours, units, cost, etc.).
  
• The coefficient 6y could indicate a rate (e.g., 6 units per time period).
  
• The term 2y adds additional proportional requirements or costs.

Combine like terms:
6y + 2y = 8y,
so the inequality becomes:
8y ≤ 90

Now, solving for y gives:
y ≤ 90 ÷ 8
y ≤ 11.25

Key Insights

Interpretation:

This inequality means y can be any value less than or equal to 11.25. In practical terms, this could represent situations such as:

• A customer buying up to 11.25 units of a product without exceeding a budget limit of $90.
  
• A worker allocating a maximum of 11.25 hours per day within an 8-hour shift constraint.
  
• A budget cap where 8y represents total spending, not to exceed $90.

Solving Linearity: Step-by-Step Guide

To solve 6y + 2y ≤ 90, follow these steps:

1. Combine like terms:
   (6 + 2)y = 8y
   So, the inequality becomes:
   8y ≤ 90

Final Thoughts

2. Isolate y:
   Divide both sides by 8 (a positive number, so the inequality direction remains unchanged):
   y ≤ 90 ÷ 8
   y ≤ 11.25

3. Interpret the solution:
   y is bounded above by 11.25; y can be zero or any positive number up to 11.25.

Real-World Applications of This Inequality

1. Budget Management

Suppose you are buying two types of items priced at $6 and $2 per unit, and your total budget is $90. If you buy 6y units of the first item and 2y units of the second, the inequality ensures spending stays within budget.

2. Time and Resource Allocation

A worker assigned tasks costing 6 minutes per unit of y and 2 minutes per auxiliary component, with a total limit of 90 minutes, must satisfy:
6y + 2y ≤ 90 → y ≤ 11.25, helping schedule work effectively.

3. Manufacturing Constraints

In production, if two resources cost $6 and $2 per batch and total cost must not exceed $90, y indicates the batch size limit for sustainable operations.