- Define the linear functions \(f(x) = 7x + 5\) and \(g(x) = 2x + 35\).
- Solve the equation \(f(x) = g(x)\) to find the intersection point \(x = 6\), which acts as the boundary between the two possible regions.
- Compare the slopes of the functions; since the slope of \(f(x)\) is greater than the slope of \(g(x)\), the function \(f(x)\) becomes larger than \(g(x)\) after the intersection point.
- The solution to the inequality \(7x + 5 > 2x + 35\) is the interval \(\boxed{x > 6}\).
Explanation
- Defining the Functions and the Relationship
To analyze the inequality \(7x + 5 > 2x + 35\), we can define two linear functions: \(f(x) = 7x + 5\) and \(g(x) = 2x + 35\). The original inequality is then equivalent to finding the range of \(x\) values for which \(f(x) > g(x)\). Both functions represent straight lines on a Cartesian plane, and their relationship depends on their slopes and their point of intersection.
- Finding the Intersection Point
The equation \(7x + 5 = 2x + 35\) represents the point where the two lines intersect. By solving this equation, we find the critical value of \(x\) that separates the domain into intervals where one function might be greater than the other. Subtracting \(2x\) from both sides gives \(5x + 5 = 35\), and subtracting 5 from both sides yields \(5x = 30\), which results in \(x = 6\). At \(x=6\), the two functions have the same value: \(f(6) = 7(6) + 5 = 47\) and \(g(6) = 2(6) + 35 = 47\).
- Analyzing the Behavior and Concluding the Solution
The slopes of our lines are the coefficients of \(x\), which are 7 for \(f(x)\) and 2 for \(g(x)\). Since \(7 > 2\), the function \(f(x)\) is steeper and increases faster than \(g(x)\). Because \(f(x)\) starts with a smaller y-intercept (\(5 < 35\)) but grows more quickly, it must be below \(g(x)\) to the left of the intersection point (\(x < 6\)) and above \(g(x)\) to the right of the intersection point (\(x > 6\)).
Examples
Imagine you are comparing two different payment plans for a service. Plan A costs a base fee of \(\$5\) plus \(\$7\) per hour, while Plan B costs a base fee of \(\$35\) plus \(\$2\) per hour. Solving the equation allows you to find the exact number of hours where both plans cost the same. Understanding the inequality helps you determine the range of hours for which one plan is cheaper than the other, enabling you to make an informed financial decision based on how many hours of service you expect to use.