The problem asks us to find the number line that correctly represents the solution to the inequality $-3x + 7 \ge 16$.

1. Problem Description

The problem asks us to find the number line that correctly represents the solution to the inequality $-3x + 7 \ge 16$.

2. Solution Steps

First, we solve the inequality for $x$:

$-3x + 7 \ge 16$

Subtract 7 from both sides:

$-3x \ge 16 - 7$

$-3x \ge 9$

Divide both sides by -

3. Remember to flip the inequality sign when dividing by a negative number:

$x \le \frac{9}{-3}$

$x \le -3$

Now, we look for the number line that represents $x \le -3$. This means the line should be shaded to the left of -3, and include -3 (closed circle or bracket at -3).

The first number line has a closed circle (or bracket) at 3 and is shaded to the right of 3 which does not represent the solution.

The second number line has a closed circle (or bracket) at 3 and is shaded to the left of 3 which does not represent the solution.

The third number line has a closed circle (or bracket) at -3 and is shaded to the left of -3 which represents the solution.

The fourth number line has an open circle (or parenthesis) at -3 and is shaded to the left of -3 which does not represent the solution.

3. Final Answer

The third number line represents the solution.