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We are tasked with solving the inequality:

\[
(u+1)(u-1)(u-6) \geq 0
\]

### Step 1: Identify the critical points

The critical points occur where each factor equals zero:
1. \( u + 1 = 0 \) gives \( u = -1 \),
2. \( u - 1 = 0 \) gives \( u = 1 \),
3. \( u - 6 = 0 \) gives \( u = 6 \).

Thus, the critical points are \( u = -1, u = 1, \) and \( u = 6 \). These are the values where the expression may change sign.

### Step 2: Test intervals

The critical points divide the real line into four intervals:

1. \( (-\infty, -1) \),
2. \( (-1, 1) \),
3. \( (1, 6) \),
4. \( (6, \infty) \).

We will test the sign of the expression in each interval by choosing a test point in each.

#### Interval \( (-\infty, -1) \):
Choose \( u = -2 \). Substituting into \( (u+1)(u-1)(u-6) \):
\[
(-2+1)(-2-1)(-2-6) = (-1)(-3)(-8) = -24.
\]
Thus, the expression is negative on \( (-\infty, -1) \).

#### Interval \( (-1, 1) \):
Choose \( u = 0 \). Substituting into \( (u+1)(u-1)(u-6) \):
\[
(0+1)(0-1)(0-6) = (1)(-1)(-6) = 6.
\]
Thus, the expression is positive on \( (-1, 1) \).

#### Interval \( (1, 6) \):
Choose \( u = 2 \). Substituting into \( (u+1)(u-1)(u-6) \):
\[
(2+1)(2-1)(2-6) = (3)(1)(-4) = -12.
\]
Thus, the expression is negative on \( (1, 6) \).

#### Interval \( (6, \infty) \):
Choose \( u = 7 \). Substituting into \( (u+1)(u-1)(u-6) \):
\[
(7+1)(7-1)(7-6) = (8)(6)(1) = 48.
\]
Thus, the expression is positive on \( (6, \infty) \).

### Step 3: Determine solution set

We want the expression to be greater than or equal to zero. Therefore, the solution set consists of the intervals where the expression is positive or zero, including the critical points where the expression equals zero.

- The expression is zero at \( u = -1, u = 1, \) and \( u = 6 \).
- The expression is positive on \( (-1, 1) \) and \( (6, \infty) \).

Thus, the solution set in interval notation is:

\[
[-1, 1] \cup [6, \infty)
\]

Would you like further details or have any questions? 

Here are 5 related questions:
1. What is the general method for solving polynomial inequalities?
2. How do you know when to include or exclude the endpoints in an inequality?
3. Can this inequality be solved using a graphing method?
4. How do you find the critical points for a product of linear factors?
5. How does the behavior of the inequality change as the degree of the polynomial increases?

**Tip**: When solving inequalities, testing values in each interval divided by critical points can help you determine the sign of the expression on those intervals.

(u+1)(u-1)(u-6) ≥ 0. Solve the inequality. Write the solution set in interval notation, if possible. Simplify the answer.

Learn how to solve the polynomial inequality (u+1)(u-1)(u-6) ≥ 0 by identifying critical points and testing intervals. This problem involves key algebraic concepts like sign analysis and the zero-product property. Suitable for high school students in Grades 9-12.

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