Question

Solve the inequality and graph the solution.
2≤
q
2
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 \( 2 \leq q^2 \), we can start by rearranging it:

1. Rewrite the inequality:
\[
q^2 \geq 2
\]

2. Take the square root of both sides. Remember to consider both the positive and negative square roots:
\[
q \geq \sqrt{2} \quad \text{or} \quad q \leq -\sqrt{2}
\]

This means that \( q \) can take any value that is either greater than or equal to \( \sqrt{2} \) or less than or equal to \( -\sqrt{2} \).

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

### Graphing the Solution

To graph the solution, we will need to:

- Plot the endpoints \( -\sqrt{2} \) and \( \sqrt{2} \).
- Since the inequality includes equality (≥), we will use closed circles at the endpoints.
- Shade the regions to the left of \( -\sqrt{2} \) and to the right of \( \sqrt{2} \).

### Completed Graph:

1. **Endpoints**:
- \( -\sqrt{2} \) (approximately -1.414) – closed circle.
- \( \sqrt{2} \) (approximately 1.414) – closed circle.

2. **Shaded Areas**:
- Shade to the left of \( -\sqrt{2} \) (negative infinity up to -\sqrt{2}).
- Shade to the right of \( \sqrt{2} \) (from \( \sqrt{2} \) to positive infinity).

The final result visually represents the solution of the compound inequality.