https://file.aiquickdraw.com/mathbot/file/17283155851987xb3er9u.jpg

The inequality in the image is:

\[
-4x \geq ax
\]

This is an inequality involving the variable \( x \), where \( a \) is a constant.

### To solve for \( x \), we can proceed as follows:

1. First, bring all the terms involving \( x \) to one side of the inequality:
   \[
   -4x - ax \geq 0
   \]
   
2. Factor out \( x \) from both terms on the left-hand side:
   \[
   x(-4 - a) \geq 0
   \]
   
3. Now, the solution depends on the sign of \( -4 - a \). We need to analyze two cases:

   - **Case 1**: If \( -4 - a > 0 \), then \( x \geq 0 \).
   - **Case 2**: If \( -4 - a < 0 \), then \( x \leq 0 \).

4. Therefore, the solution depends on the value of \( a \). If \( a = -4 \), the inequality becomes \( 0 \geq 0 \), which is true for any \( x \), meaning the solution is all real numbers.

Would you like more details on any of these steps?

Here are 5 related questions to further expand:

1. What happens if \( a \) is positive or negative in this inequality?
2. How would the inequality change if it were \( > \) instead of \( \geq \)?
3. What are the implications if \( a = -4 \)?
4. How does factoring play a role in solving inequalities?
5. Could this inequality have no solution for certain values of \( a \)?

### Tip:
When solving inequalities, always pay attention to how the direction of the inequality might change when multiplying or dividing by a negative number.

Solve the inequality -4x ≥ ax, where a is a constant.

Learn how to solve the inequality -4x ≥ ax involving the constant 'a' and variable 'x'. This step-by-step solution covers key algebraic techniques such as factoring and sign analysis of inequalities. Suitable for students in Grades 9-12.

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