Solve the inequality and graph the solution.

Question

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.