- Define the functions \(f(x) = 7x + 5\) and \(g(x) = 2x + 35\) and note that their intersection, \(x=6\), marks the boundary for the inequality.
- Introduce the difference function \(h(x) = f(x) - g(x) = 5x - 30\), where the inequality \(f(x) > g(x)\) becomes \(h(x) > 0\).
- Recognize that \(h(x)\) is an increasing function due to its positive slope, meaning \(h(x) > 0\) exclusively for \(x > 6\).
- Conclude that the intersection point \(x=6\) determines the threshold, and the solution to the inequality is \(\boxed{x > 6}\).
Explanation
- Analyzing the Functions and the Equation
We are considering the linear inequality \(7x + 5 > 2x + 35\). Let us define two functions, \(f(x) = 7x + 5\) and \(g(x) = 2x + 35\). Solving the equation \(f(x) = g(x)\) means finding the value of \(x\) where the two lines intersect. By subtracting \(2x\) and \(5\) from both sides of the equation, we get \(5x = 30\), which gives the intersection point at \(x = 6\). This value serves as the critical point that separates the number line into intervals where one function might be greater than the other.
- The Role of the Intersection Point as a Boundary
To understand the relationship between the two functions, we can define a new function \(h(x) = f(x) - g(x) = (7x + 5) - (2x + 35) = 5x - 30\). The original inequality \(7x + 5 > 2x + 35\) is equivalent to \(h(x) > 0\). The equation we solved, \(7x + 5 = 2x + 35\), is equivalent to \(h(x) = 0\), which occurs at \(x = 6\). The function \(h(x) = 5x - 30\) is a linear function with a slope of 5. Since the slope is positive, \(h(x)\) is a strictly increasing function.
- Determining the Solution Interval
Because \(h(x)\) is a strictly increasing linear function that equals 0 at \(x = 6\), we know that for any value of \(x\) greater than 6, \(h(x)\) must be greater than 0, and for any value of \(x\) less than 6, \(h(x)\) must be less than 0. This confirms that the inequality \(h(x) > 0\) is satisfied exactly when \(x > 6\). Thus, the solution to the equation provides the boundary point necessary to identify the solution interval of the inequality.
Examples
Understanding how the root of an equation acts as a boundary for an inequality is useful in many real-life optimization problems. For example, if you are comparing two pricing models, say Model A costing \(\$ (7x + 5)\) and Model B costing \(\$ (2x + 35)\), finding the intersection point \(x = 6\) tells you the exact usage level where Model B becomes cheaper than Model A. Knowing this "break-even" point allows you to make informed decisions about which option to choose depending on your usage.