Math Problem Statement

A graph is shown, and the user is asked to solve f(x) > 0, which means finding the intervals where the graph is above the x-axis. The user is also asked to solve f(x) < 0, meaning where the graph is below the x-axis.

Solution

Let's analyze this graph and the questions provided.

Part A) Understanding f(x)>0f(x) > 0

In the graph, f(x)>0f(x) > 0 means the values where the graph is above the x-axis (the line y=0y = 0).

So, the correct choice is "above the x-axis".

Part B) Finding values of xx where f(x)>0f(x) > 0

To answer this part, we need to determine where the graph lies above the x-axis:

  • The graph is positive (above the x-axis) in two regions:
    • On the left side, starting from x=x = -\infty and stopping at x5.5x \approx -5.5.
    • On the right side, for x>2x > 2.

Thus, the solution is: (,5.5)(2,)(-\infty, -5.5) \cup (2, \infty)

Part C) Finding values of xx where f(x)<0f(x) < 0

Here, f(x)<0f(x) < 0 means finding where the graph is below the x-axis. Observing the graph, this happens between:

  • x5.5x \approx -5.5 and x=2x = 2.

Thus, the interval is: (5.5,2)(-5.5, 2)


Do you need more detailed explanations, or have any questions on this so far?

Here are 5 follow-up questions to deepen understanding:

  1. Why is f(x)=0f(x) = 0 at the points where the graph touches the x-axis?
  2. How do you interpret f(x)f(x) in terms of real-world applications, such as physics or economics?
  3. How would you write the solution to f(x)=0f(x) = 0?
  4. Can you estimate the exact coordinates where the function crosses the x-axis?
  5. How does the behavior of the graph at the asymptotes influence the intervals?

Tip: When solving inequalities involving functions, always check the graph’s interaction with the x-axis to determine positive and negative regions.

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Math Problem Analysis

Mathematical Concepts

Graphing
Inequalities
Function Analysis

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12