Let's solve the compound inequality step by step:

### Given:
\[
-3 < 1 - 6x \leq 4
\]

### Step 1: Split the compound inequality
We can split it into two parts:

1. \(-3 < 1 - 6x\)
2. \(1 - 6x \leq 4\)

### Step 2: Solve each inequality

#### First part: 
\(-3 < 1 - 6x\)

Subtract \(1\) from both sides:
\[
-3 - 1 < -6x
\]
\[
-4 < -6x
\]

Now, divide by \(-6\) (remember to flip the inequality sign when dividing by a negative number):
\[
\frac{-4}{-6} > x
\]
\[
\frac{2}{3} > x
\]
or equivalently:
\[
x < \frac{2}{3}
\]

#### Second part:
\(1 - 6x \leq 4\)

Subtract \(1\) from both sides:
\[
-6x \leq 3
\]

Now, divide by \(-6\) (again, flip the inequality sign):
\[
x \geq \frac{3}{-6}
\]
\[
x \geq -\frac{1}{2}
\]

### Step 3: Combine the results

The solution to the compound inequality is:
\[
-\frac{1}{2} \leq x < \frac{2}{3}
\]

This means \(x\) is in the range between \(-\frac{1}{2}\) and \(\frac{2}{3}\), inclusive of \(-\frac{1}{2}\) but not \(\frac{2}{3}\).

Would you like any further explanation or details?

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Here are five related questions to expand your understanding:

1. How do you solve compound inequalities involving both strict and non-strict inequalities?
2. What happens if the inequality involves absolute values?
3. Can you graph the solution \(-\frac{1}{2} \leq x < \frac{2}{3}\) on a number line?
4. How does solving an inequality differ from solving an equation?
5. What are the rules for flipping inequality signs when manipulating inequalities?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Explore the step-by-step solution for the compound inequality 3 < 1 - 6x <= 4. This problem delves into core algebraic concepts, including inequality solving techniques and key rules for handling negative coefficients. Perfect for students in Grades 7-9.

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