Speeding Fines Suppose that speeding fines are determined by...

Speeding Fines Suppose that speeding fines are determined by $y=10(x-65)+50, x>65,$ where $y$ is the cost in dollars of the fine if a person is caught driving $x$ miles per hour. (a) How much is the fine for driving 76 mph? (b) While balancing the checkbook, Johnny found a check that his wife Gwen had written to the Department of Motor Vehicles for a speeding fine. The check was written for $\$ 100 .$ How fast was Gwen driving? (c) At what whole-number speed are tickets first given? (d) For what speeds is the fine greater than $\$ 200 ?$

Speeding Fines Suppose that speeding fines are determined by $y=10(x-65)+50, x>65,$ where $y$ is the cost in dollars of the fine if a person is caught driving $x$ miles per hour.
(a) How much is the fine for driving 76 mph?
(b) While balancing the checkbook, Johnny found a check that his wife Gwen had written to the Department of Motor Vehicles for a speeding fine. The check was written for $\$ 100 .$ How fast was Gwen driving?
(c) At what whole-number speed are tickets first given?
(d) For what speeds is the fine greater than $\$ 200 ?$

Key Concepts

Inequalities in Linear Equations

Inequalities extend the concept of equations to include a range of values, rather than a single value. They are used to determine conditions like when the fine exceeds a certain amount, helping analyze scenarios where the output of the function is bounded by a threshold.

Inverse Functions

The idea of an inverse function involves reversing the roles of the input and output. In this context, once a fine is calculated from the speed, working backwards by solving the equation helps determine the speed based on a known fine amount.

Linear Functions

A linear function represents a relationship where the change in the output is directly proportional to the change in the input, typically expressed in the form y = mx + b. This concept is fundamental in modeling real-world situations like speeding fines, where the rate of increase in fines is constant above a certain speed.

Domain Restrictions

In many function models, the function is only applicable to values within a specific range, known as the domain. In the context of speeding fines, the fine applies only when the speed exceeds a particular limit, demonstrating the importance of considering the valid input values for the function.

Solving Linear Equations

Solving a linear equation involves finding the value of the variable that satisfies the equation. This process is used directly when determining the fine for a given speed or finding the speed corresponding to a specific fine.

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