Solve the inequality and graph the solution.

Question

f
2
≥3
Plot the endpoints. Select an endpoint to change it from closed to open. Select the middle of the segment, ray, or line to delete it.

Answer 1

To solve the inequality \( f^2 \geq 3 \), we can begin by isolating \( f \).

1. Start by taking the square root of both sides, remembering to consider both the positive and negative roots: \[ f \geq \sqrt{3} \quad \text{or} \quad f \leq -\sqrt{3} \]

This means \( f \) can be any value greater than or equal to \( \sqrt{3} \) or less than or equal to \( -\sqrt{3} \).

2. The solution can be expressed in interval notation as: \[ (-\infty, -\sqrt{3}] \cup [\sqrt{3}, \infty) \]

Graphing the solution

To graph the solution, we'll create a number line:

1. Endpoints: Place the points \( -\sqrt{3} \) and \( \sqrt{3} \) on the number line.
2. Closed intervals: Since the endpoints are included when the inequality is \( \geq \) or \( \leq \), we will use closed dots at these points.
3. Ray representation:
   • Shade or draw arrows to the left from \( -\sqrt{3} \) to include all numbers less than or equal to \( -\sqrt{3} \).
   • Shade or draw arrows to the right from \( \sqrt{3} \) to include all numbers greater than or equal to \( \sqrt{3} \).

Example graph:

Below is a simplified representation of what the graph would look like on a number line:

<---•===================•------>
     -√3             √3

• The dot at \( -\sqrt{3} \) is closed, indicating \( -\sqrt{3} \) is included.
• The dot at \( \sqrt{3} \) is also closed, indicating \( \sqrt{3} \) is included.
• The arrows indicate that the shading continues indefinitely in both directions.

Select an endpoint to change:

If you need to choose one of the endpoints to change from closed to open (for example, you wish to exclude one endpoint from the solution), you might choose \( \sqrt{3} \). If you do this, the new solution would be:

1. The new intervals would be: \[ (-\infty, -\sqrt{3}] \cup (-\sqrt{3}, \sqrt{3}) \]

New Graph with Changed Endpoint:

<---•==================o------>
     -√3             √3

• The dot at \( -\sqrt{3} \) is still closed.
• The dot at \( \sqrt{3} \) is now open, indicating \( \sqrt{3} \) is not included in the solution.
• The arrow to the right side still reflects that \( f \) can take values greater than \( \sqrt{3} \).