You are given that a function f is defined and continuous at every real number x except x=2. Also, f(x)=0 precisely for x=1 and 7. Finally, you know that f(-3)=4, f(1.5)=2, f(4)= -3, and f(10)=6. Solve the inequality f(x) > 0.
To solve the inequality f(x) > 0, we need to determine the intervals where the function f(x) is greater than zero.
Given that f(x) is continuous at every real number except x = 2, we can break down the problem into smaller intervals based on the known points where f(x) crosses the x-axis (f(x) = 0). In this case, we have f(x) = 0 at x = 1 and x = 7.
1. Firstly, let's consider the interval (-∞, 1).
Since f(x) = 0 at x = 1, we know that f(x) changes sign in this interval. To determine whether f(x) is positive or negative in this interval, we can choose any point within this interval and evaluate the function.
Let's choose x = 0 as a test point in the interval (-∞, 1).
f(0) = -3
Since f(0) = -3, which is negative, we can conclude that f(x) is negative for x < 1 in the interval (-∞, 1).
2. Next, let's consider the interval (1, 7).
Since f(x) = 0 at x = 1 and x = 7, we know that f(x) changes sign in this interval as well. To determine whether f(x) is positive or negative in this interval, we can again choose any test point within this interval and evaluate the function.
Let's choose x = 5 as a test point in the interval (1, 7).
f(5) = -3
Since f(5) = -3, which is negative, we can conclude that f(x) is negative for 1 < x < 7 in the interval (1, 7).
3. Lastly, let's consider the interval (7, ∞).
Since we do not have any additional information about the function f(x) after x = 7, we cannot determine whether f(x) is positive or negative in this interval.
Therefore, the solution to the inequality f(x) > 0 is the union of the intervals where f(x) is positive:
(-∞, 1) ∪ (7, ∞)