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The graph of \( y = f(x) \) is given, and we are tasked with solving inequalities based on this graph. Specifically, we want to determine where the function \( f(x) \) is positive (above the x-axis) and negative (below the x-axis). The solutions should be written in interval notation.

### Step-by-Step Breakdown:

1. **Find the x-intercepts (zeros of the function):** These are the points where \( f(x) = 0 \), meaning the function crosses the x-axis. From the graph, the approximate x-intercepts are:
   - \( x = -4 \)
   - \( x = 2 \)
   - \( x = 5 \)

2. **Identify intervals where \( f(x) \) is positive or negative:**
   - The function is **positive** where the graph is above the x-axis.
   - The function is **negative** where the graph is below the x-axis.

3. **Intervals of positivity (\( f(x) > 0 \)):**
   - From \( x \in (-4, 2) \), the graph is below the x-axis, so there is no positivity in this region.
   - The function is positive between \( x = 2 \) and \( x = 5 \), meaning \( f(x) > 0 \) on the interval:
     \[
     (2, 5)
     \]

4. **Intervals of negativity (\( f(x) < 0 \)):**
   - For \( x \in (-\infty, -4) \), the function is negative (graph is below the x-axis).
   - Between \( x = -4 \) and \( x = 2 \), the function remains negative.
   - Finally, for \( x \in (5, \infty) \), the graph again dips below the x-axis.

   So, the function is negative in the intervals:
   \[
   (-\infty, -4) \cup (-4, 2) \cup (5, \infty)
   \]

### Solution Sets in Interval Notation:

- \( f(x) > 0 \) on the interval \( (2, 5) \).
- \( f(x) < 0 \) on the intervals \( (-\infty, -4) \cup (-4, 2) \cup (5, \infty) \).

Do you have any questions about these inequalities or need more details?

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Here are 5 related questions that may help you expand your understanding:

1. How do you determine if a function is positive or negative based solely on its graph?
2. What happens at the x-intercepts of a function in terms of positivity or negativity?
3. Can a function have more than one interval where it is positive or negative?
4. How would the solution change if the graph touched the x-axis but didn’t cross it at certain points?
5. How would you find the exact values of the x-intercepts algebraically?

**Tip:** Always pay attention to the behavior of the function around the x-intercepts when solving inequalities; the function switches sign at these points.

The graph of y = f(x) is given. Solve the inequalities. Write the solution sets in interval notation.

Solve inequalities based on the graph of y = f(x), identifying intervals where the function is positive or negative. Learn how to express solutions in interval notation.

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