To solve the inequality \( h^2 \geq 2 \), we start by manipulating the inequality so that we can find the values of \( h \).
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Isolate the square term:
\[ h^2 \geq 2 \]
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Take the square root of both sides. Remember that when dealing with inequalities, we need to consider both the positive and negative roots:
\[ h \geq \sqrt{2} \quad \text{or} \quad h \leq -\sqrt{2} \]
Thus, the solutions to the inequality \( h^2 \geq 2 \) are \( h \leq -\sqrt{2} \) or \( h \geq \sqrt{2} \).
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Endpoints of the intervals: The endpoints of the intervals are \( -\sqrt{2} \) and \( \sqrt{2} \).
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Graph the solution:
- Draw a number line.
- Mark the points \( -\sqrt{2} \) and \( \sqrt{2} \).
- Since the inequality is \( \geq \) (greater than or equal to) and \( \leq \) (less than or equal to), we will use closed circles on both \( -\sqrt{2} \) and \( \sqrt{2} \).
- Shade the region to the left of \( -\sqrt{2} \) and to the right of \( \sqrt{2} \).
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Choose one endpoint to change to open: For example, we can choose \( \sqrt{2} \) to be open. This would change the inequality to \( h \geq -\sqrt{2} \) but leave \( h < \sqrt{2} \) with an open circle at \( \sqrt{2} \).
Here’s how the graph will look:
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Endpoints:
- Closed circle at \( -\sqrt{2} \) (since it can be equal).
- Open circle at \( \sqrt{2} \) (since it cannot be equal).
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Shaded region: The line extends to the left from \( -\sqrt{2} \) and goes to negative infinity, and the line extends to the right from \( \sqrt{2} \) to positive infinity.
Conclusion
The solution, represented graphically, indicates that \( h \) can be any value less than or equal to \( -\sqrt{2} \) or any value greater than \( \sqrt{2} \) (excluding \( \sqrt{2} \) itself).