To solve the compound inequality:

\[
-3 < 1 - 6x \leq 4
\]

We will break it down into two parts and solve step by step.

### Part 1: Solve \( -3 < 1 - 6x \)

1. Subtract 1 from both sides:

   \[
   -3 - 1 < -6x
   \]

   \[
   -4 < -6x
   \]

2. Divide both sides by -6. When dividing by a negative number, remember to reverse the inequality sign:

   \[
   \frac{-4}{-6} > x
   \]

   \[
   \frac{2}{3} > x \quad \text{or} \quad x < \frac{2}{3}
   \]

### Part 2: Solve \( 1 - 6x \leq 4 \)

1. Subtract 1 from both sides:

   \[
   -6x \leq 4 - 1
   \]

   \[
   -6x \leq 3
   \]

2. Divide both sides by -6, and reverse the inequality:

   \[
   x \geq \frac{3}{-6}
   \]

   \[
   x \geq -\frac{1}{2}
   \]

### Final Solution:

We combine both results:

\[
-\frac{1}{2} \leq x < \frac{2}{3}
\]

This is the solution to the compound inequality.

Would you like more details, or have any questions?

Here are 5 related questions:

1. How do you solve compound inequalities involving three parts?
2. What happens when you divide by a negative number in inequalities?
3. Can you represent the solution on a number line?
4. How would you check if a number is part of the solution set?
5. How do you solve compound inequalities with absolute values?

**Tip:** Always reverse the inequality when multiplying or dividing by a negative number.

Learn how to solve the compound inequality 3 < 1 - 6x <= 4 step by step. This guide covers algebraic techniques, including handling compound inequalities and reversing the inequality when dividing by negative numbers. Ideal for students in Grades 8-10.

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