To solve the inequality \( x^2 < 3 \), we start by rewriting it as:
\[ x^2 - 3 < 0 \]
Next, we find the points where \( x^2 - 3 = 0 \) by solving:
\[ x^2 = 3 \]
Taking the square root of both sides, we have:
\[ x = \sqrt{3} \quad \text{and} \quad x = -\sqrt{3} \]
These points, \( -\sqrt{3} \) and \( \sqrt{3} \), divide the number line into three intervals:
- \( (-\infty, -\sqrt{3}) \)
- \( (-\sqrt{3}, \sqrt{3}) \)
- \( (\sqrt{3}, \infty) \)
To determine which intervals satisfy the inequality \( x^2 < 3 \), we can test a value from each interval:
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For \( x = -2 \) (in the interval \( (-\infty, -\sqrt{3}) \)): \[ (-2)^2 = 4 \quad (\text{not a solution, since } 4 \nless 3) \]
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For \( x = 0 \) (in the interval \( (-\sqrt{3}, \sqrt{3}) \)): \[ 0^2 = 0 \quad (\text{solution, since } 0 < 3) \]
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For \( x = 2 \) (in the interval \( (\sqrt{3}, \infty) \)): \[ 2^2 = 4 \quad (\text{not a solution, since } 4 \nless 3) \]
Thus, the solution to the inequality \( x^2 < 3 \) is:
\[ -\sqrt{3} < x < \sqrt{3} \]
On a number line, this is represented by a line segment between \( -\sqrt{3} \) and \( \sqrt{3} \), with open endpoints (indicating that \( -\sqrt{3} \) and \( \sqrt{3} \) are not included in the solution).
If you have a specific number line image or representation to refer to with your question, please provide it, and I can help you identify the correct one.
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