To solve the inequality:

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

We can break it into two parts and solve them separately.

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

1. Subtract \(1\) from both sides:

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

2. Divide both sides by \(-6\) (remember to reverse the inequality sign because we are dividing by a negative number):

\[
\frac{-4}{-6} > x
\]
\[
\frac{2}{3} > x \quad \text{or} \quad x < \frac{2}{3}
\]

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

1. Subtract \(1\) from both sides:

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

2. Divide both sides by \(-6\) (again, reverse the inequality sign):

\[
x \geq \frac{3}{-6}
\]
\[
x \geq -\frac{1}{2}
\]

### Final solution:
Combining both inequalities, we get:

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

The solution is:

\[
x \in \left[ -\frac{1}{2}, \frac{2}{3} \right)
\]

This means \(x\) is greater than or equal to \(-\frac{1}{2}\) and strictly less than \(\frac{2}{3}\).

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How would the solution change if the inequality were strict on both sides?
2. What is the difference between solving a compound inequality and solving two separate inequalities?
3. How do you graph the solution \(\left[ -\frac{1}{2}, \frac{2}{3} \right)\) on a number line?
4. What happens when both inequalities involve multiplying by a negative number?
5. How do you handle inequalities when there are more variables involved?

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

Learn how to solve the compound inequality 3 < 1 - 6x <= 4 with step-by-step instructions. This problem covers key algebraic concepts, such as solving inequalities with negative coefficients and handling compound inequalities. Suitable for students in Grades 8-10.

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