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Inequalities are mathematical expressions showing that one quantity is greater than, less than, or not equal to another. They are crucial in many real-world scenarios, such as deciding how many products need to be sold to break even or make a profit. Consider the inequality in this exercise: the revenue must be greater than the costs for a profit to occur. Inequality symbols include:

• ">" (greater than)
• "<" (less than)
• "≥" (greater than or equal to)
• "≤" (less than or equal to)

When working with inequalities, remember that performing certain operations, like multiplying or dividing both sides by a negative number, reverses the inequality sign. Here, dividing both sides of the inequality \(-6x > 120\) by \(-6\) flipped the sign, changing the inequality from an invalid scenario, \(x > -20\), to determine where the original inequality holds true. However, real-world constraints, like a non-negative amount of products, ensure certain answers, like non-existent negative production solutions, are logically interpreted and applied.

Linear functions are mathematical expressions that create straight lines when graphed. They have the general formula \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this exercise, the cost function \(C(x) = 11x + 120\) and the revenue function \(R(x) = 5x\), are both linear.
The slope \(m\) represents the rate of change. For cost, \(m = 11\) means that the cost increases by 11 units per product. For revenue, \(m = 5\) indicates a 5-unit revenue increase per product unit sold. The y-intercept \(b\), which is \(120\) in \(C(x)\), shows fixed costs — expense at the outset before selling any product.
Understanding these concepts helps in visualizing the behavior of both functions. By comparing, one can find solution points like the break-even or profit points by setting these equal or using inequalities as shown in this exercise.

Cost and revenue analysis is essential to determining when a business becomes profitable. Costs include fixed costs, incurred regardless of production, and variable costs, changing with production level. In our scenario, \(C(x) = 11x + 120\) highlights this, with \(11x\) as the variable part and \(120\) as fixed.Revenue refers to the total money a business makes, expressed as \(R(x) = 5x\). The goal is to find when this revenue exceeds costs, as seen in the inequality \(R(x) > C(x)\) or \(5x > 11x + 120\). In a typical analysis, finding when these lines intersect (break-even) or when revenue surpasses cost (profit) marks critical decision points.
Yet in this case, the inequality solution, \(x < -20\), proved technically infeasible, indicating a fundamental issue with pricing or cost structure needing reassessment. Thus, cost and revenue analysis not only identifies potential profit points but also highlights when business strategies need re-evaluation. By understanding these mathematical models, businesses can better prepare for financial success.