https://file.aiquickdraw.com/mathbot/file/1727635921506zysfdqxz.jpg

The problem presented is the polynomial inequality:

\[
(x - 1)^2 \geq 4
\]

### Step 1: Isolate the expression.
Start by solving the corresponding equality to find the critical points:

\[
(x - 1)^2 = 4
\]

Take the square root of both sides:

\[
x - 1 = \pm 2
\]

Thus, the two solutions are:

\[
x = 1 + 2 = 3 \quad \text{or} \quad x = 1 - 2 = -1
\]

So the critical points are \(x = -1\) and \(x = 3\).

### Step 2: Solve the inequality.
Now we solve the inequality:

\[
(x - 1)^2 \geq 4
\]

This inequality is satisfied when \((x - 1)^2 = 4\), which we already found as \(x = -1\) and \(x = 3\). Additionally, since \((x - 1)^2\) represents a parabola opening upwards, it is greater than or equal to 4 outside the interval \((-1, 3)\). So, we test values in the intervals:

1. **For \(x < -1\)** (e.g., \(x = -2\)):
   \[
   (-2 - 1)^2 = 9 \geq 4
   \]
   This is true, so \(x < -1\) satisfies the inequality.

2. **For \(-1 < x < 3\)** (e.g., \(x = 0\)):
   \[
   (0 - 1)^2 = 1 \not\geq 4
   \]
   This is false, so the inequality does not hold in this interval.

3. **For \(x > 3\)** (e.g., \(x = 4\)):
   \[
   (4 - 1)^2 = 9 \geq 4
   \]
   This is true, so \(x > 3\) satisfies the inequality.

### Step 3: Interval notation.
The solution includes \(x = -1\) and \(x = 3\) because the inequality is "greater than or equal to." Therefore, the solution in interval notation is:

\[
(-\infty, -1] \cup [3, \infty)
\]

This is the set of all \(x\) values that satisfy the inequality.

---

Would you like further details or have any questions?

Here are some related questions to deepen understanding:
1. How do you determine when to use a union of intervals in inequalities?
2. What is the graphical representation of \( (x - 1)^2 = 4 \)?
3. Why do we need to check each interval between critical points for inequalities?
4. What happens if the inequality were strict, \( (x - 1)^2 > 4 \)?
5. How do you solve inequalities with absolute values?

**Tip:** When solving polynomial inequalities, always solve the related equation first to find critical points and then check the intervals around those points.

Learn how to solve the polynomial inequality (x - 1)^2 ≥ 4 using algebraic methods. This solution covers key concepts in inequalities and polynomials, suitable for students in Grades 9-10. Includes a detailed step-by-step solution and interval notation.

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