I don't get this part, "This inequality holds when:

x≤−1orx≥1*x*≤−1or*x*≥1

"/ How did the inequality switch for x<=-1

Great question! Let's walk through that step in more detail.

We start with the inequality:

\[
x^2 \geq 1
\]

### Step 1: Understanding \( x^2 \geq 1 \)
This inequality says that the square of \( x \) must be greater than or equal to 1. To understand when this is true, let's break it down:

- **Square Root Analysis**: \( x^2 = 1 \) has two solutions: \( x = 1 \) and \( x = -1 \).
- **Greater Than Analysis**: For \( x^2 \) to be greater than 1, \( x \) must either be larger than 1 or smaller than -1. 

### Step 2: Solve for \( x \)
To solve the inequality \( x^2 \geq 1 \), let's consider both possible cases:

1. **When \( x \geq 0 \)**:
   \[
   x^2 \geq 1 \implies x \geq 1
   \]
   Here, \( x \) must be greater than or equal to 1.

2. **When \( x < 0 \)**:
   \[
   x^2 \geq 1 \implies x \leq -1
   \]
   Here, \( x \) must be less than or equal to -1.

### Step 3: Combine the Results
The solution to the inequality \( x^2 \geq 1 \) is the union of these two intervals:

- \( x \geq 1 \) or \( x \leq -1 \)

In interval notation, this is expressed as:

\[
(-\infty, -1] \cup [1, \infty)
\]

### Summary
The reason the inequality \( x^2 \geq 1 \) leads to \( x \leq -1 \) or \( x \geq 1 \) is because for the square of \( x \) to be greater than or equal to 1, \( x \) must either be sufficiently negative (less than or equal to -1) or sufficiently positive (greater than or equal to 1). 

If you have any further questions or need clarification, feel free to ask! Here are some related questions you might explore:

1. What are the solutions to the inequality \( x^2 < 4 \)?
2. How does the inequality \( x^2 \leq 9 \) differ from \( x^2 \geq 9 \) in terms of solutions?
3. Can you find the domain of the function \( f(x) = \sqrt{4 - x^2} \)?
4. What would be the domain of \( g(x) = \sqrt{x^4 - 16} \)?
5. How do absolute value inequalities relate to these kinds of problems?

**Tip**: Remember that when solving inequalities involving squares, the sign of \( x \) is crucial for determining the correct solution intervals.

Learn how to understand and solve the inequality x^2 ≥ 1, exploring when the inequality switches for x ≤ -1 or x ≥ 1. This problem delves into inequalities and quadratic equations, suitable for Grades 9-12.

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