The problem asks to solve the compound inequality $-1 \le -2x + 3 < 9$ and graph the solution on the number line.

1. Problem Description

The problem asks to solve the compound inequality $-1 \le -2x + 3 < 9$ and graph the solution on the number line.

2. Solution Steps

First, we need to isolate $x$ in the inequality $-1 \le -2x + 3 < 9$. We can do this by subtracting 3 from all parts of the inequality:

$-1 - 3 \le -2x + 3 - 3 < 9 - 3$

$-4 \le -2x < 6$

Now, we divide all parts of the inequality by $-2$. Remember that when we divide or multiply by a negative number, we need to reverse the inequality signs:

$\frac{-4}{-2} \ge \frac{-2x}{-2} > \frac{6}{-2}$

$2 \ge x > -3$

We can rewrite this as:

$-3 < x \le 2$

This means $x$ is greater than $-3$ and less than or equal to $2$.

To graph the solution on the number line, we use an open circle at $-3$ because $x$ is strictly greater than $-3$, and a closed circle at $2$ because $x$ is less than or equal to $2$. We draw a line connecting these two points.

3. Final Answer

The solution to the inequality is $-3 < x \le 2$.

On the provided number line, draw an open circle at $-3$ and a closed circle at $2$, and draw a line segment connecting the two circles.