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We are tasked with solving the inequality \( 1 - 6x \leq 1 \) and determining which of the given values (I: \(-6\), II: 4, and III: 0) are solutions.

### Step 1: Solve the inequality
Start with the inequality:

\[
1 - 6x \leq 1
\]

Subtract 1 from both sides:

\[
-6x \leq 0
\]

Next, divide both sides by \(-6\). Since we're dividing by a negative number, we must reverse the inequality:

\[
x \geq 0
\]

### Step 2: Analyze the given values
- **I. \( x = -6 \)**: 
  Substituting \( x = -6 \) into the inequality:
  \[
  1 - 6(-6) = 1 + 36 = 37
  \]
  This is not less than or equal to 1, so \( x = -6 \) is **not** a solution.

- **II. \( x = 4 \)**:
  Substituting \( x = 4 \):
  \[
  1 - 6(4) = 1 - 24 = -23
  \]
  This satisfies \( -23 \leq 1 \), so \( x = 4 \) is a solution.

- **III. \( x = 0 \)**:
  Substituting \( x = 0 \):
  \[
  1 - 6(0) = 1
  \]
  This satisfies \( 1 \leq 1 \), so \( x = 0 \) is also a solution.

### Step 3: Conclusion
The values that satisfy the inequality are \( x = 4 \) and \( x = 0 \). Therefore, the correct answer is:

\[
\text{II and III}
\]

Would you like further details or explanations? 

Here are 5 related questions to expand your understanding:

1. How do you solve inequalities with negative coefficients?
2. Why must the inequality sign be reversed when dividing by a negative number?
3. Can the solution set of the inequality be represented on a number line?
4. What would happen if the inequality were strict, i.e., \(1 - 6x < 1\)?
5. How would the inequality change if it were \(1 - 6x \geq 1\)?

**Tip**: Always be cautious when multiplying or dividing an inequality by a negative number. The inequality direction reverses in such cases!

Which of the following values are solutions to the inequality 1 - 6x ≤ 1? I. -6 II. 4 III. 0

Learn how to solve the inequality 1 - 6x ≤ 1 and determine which of the values -6, 4, and 0 are solutions. This problem focuses on understanding linear inequalities and the rules of inequalities. Suitable for Grades 6-8 students.

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