Solve the inequality in terms of intervals. (Enter your answer using interval notation.)

x^2 ≥ 3

To solve the inequality \( x^2 \geq 3 \), we'll proceed with the following steps:

### 1. **Rewrite the Inequality**:
   \[
   x^2 \geq 3
   \]
   This can be rewritten as:
   \[
   x^2 - 3 \geq 0
   \]

### 2. **Find the Roots**:
   First, solve the equation \( x^2 - 3 = 0 \) to find the critical points (roots).
   \[
   x^2 = 3
   \]
   Taking the square root of both sides:
   \[
   x = \pm \sqrt{3}
   \]
   So, the critical points are \( x = \sqrt{3} \) and \( x = -\sqrt{3} \).

### 3. **Determine the Sign of the Expression**:
   To determine where the expression \( x^2 - 3 \) is positive or negative, consider the intervals defined by the critical points:
   - \( (-\infty, -\sqrt{3}) \)
   - \( (-\sqrt{3}, \sqrt{3}) \)
   - \( (\sqrt{3}, \infty) \)

   Test the sign of the expression in each interval:
   - For \( x \in (-\infty, -\sqrt{3}) \): Choose \( x = -2 \)
     \[
     (-2)^2 - 3 = 4 - 3 = 1 \geq 0
     \]
     The expression is positive.
   - For \( x \in (-\sqrt{3}, \sqrt{3}) \): Choose \( x = 0 \)
     \[
     0^2 - 3 = -3 < 0
     \]
     The expression is negative.
   - For \( x \in (\sqrt{3}, \infty) \): Choose \( x = 2 \)
     \[
     2^2 - 3 = 4 - 3 = 1 \geq 0
     \]
     The expression is positive.

### 4. **Write the Solution in Interval Notation**:
   The inequality \( x^2 \geq 3 \) is satisfied when \( x \in (-\infty, -\sqrt{3}] \) or \( x \in [\sqrt{3}, \infty) \).

Thus, the solution in interval notation is:
\[
(-\infty, -\sqrt{3}] \cup [\sqrt{3}, \infty)
\]

Would you like further details or have any questions about this solution?

Here are some related questions:
1. How do you solve \(x^2 > 4\) using interval notation?
2. What is the process for solving \(x^2 < 2\)?
3. How do you graph the solution set for \(x^2 \geq 5\)?
4. What are the key steps to find intervals for quadratic inequalities?
5. Can you explain why the inequality flips when dividing by a negative number?

**Tip:** When solving inequalities, always consider testing points within each interval to determine the sign of the expression.

Learn how to solve the inequality x^2 ≥ 3 using interval notation with a detailed step-by-step solution. This problem covers key concepts in inequalities and quadratic equations. Suitable for high school students in Grades 9-12.

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