To solve the inequality \( 2 \leq q^2 \), we can start by rearranging it:
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Rewrite the inequality: \[ q^2 \geq 2 \]
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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 \( -\sqrt{2} \).
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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:
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Endpoints:
- \( -\sqrt{2} \) (approximately -1.414) – closed circle.
- \( \sqrt{2} \) (approximately 1.414) – closed circle.
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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.