To represent the inequality $-\frac{1}{4}x \geq 4$ on a number line, follow these steps to solve it and then illustrate the solution:

Solving the Inequality

1. Isolate $x$: Multiply both sides by $-4$ to cancel out the $-\frac{1}{4}$ coefficient. Remember, when you multiply or divide by a negative number, you must reverse the inequality sign.

   $-\frac{1}{4}x \geq 4 \implies x \leq -16$

Representing on a Number Line

1. Draw the Number Line: Draw a horizontal line and mark points at regular intervals.

2. Identify the Point: Locate $-16$ on the number line.

3. Shade the Interval: Since $x \leq -16$, you will shade all points to the left of $-16$, including $-16$ itself. This is represented with a closed circle at $-16$ and a line extending indefinitely to the left.

Summary

The solution to the inequality $-\frac{1}{4}x \geq 4$ is $x \leq -16$. On a number line, this is shown by a closed circle at $-16$ and a line extending to the left.

If you have any questions or need further clarification, feel free to ask!

Here are some related questions to deepen your understanding:

1. How do you solve and graph linear inequalities with two variables?
2. What are the rules for solving inequalities involving absolute values?
3. How does one solve systems of inequalities?
4. What are the implications of multiplying or dividing inequalities by negative numbers?
5. How do you solve inequalities involving quadratic expressions?
6. How can interval notation be used to express the solution of inequalities?
7. How does one graph inequalities on the Cartesian plane?
8. What are compound inequalities, and how are they solved?

Tip: Always pay attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers.