We are given the inequality:

\[
|5 - 2x| \leq -3x
\]

To solve it, we can break it down into two cases, based on the definition of absolute value. However, before that, let's analyze the inequality:

The absolute value of any expression is always non-negative (i.e., \( |5 - 2x| \geq 0 \)). Since the right-hand side, \(-3x\), can be negative or zero, we should first check whether the inequality can even hold.

Let's consider the cases:

1. **If \( -3x \) is positive** (which happens when \( x < 0 \)), we could proceed with solving both cases of the absolute value.
2. **If \( -3x \) is negative or zero** (which happens when \( x \geq 0 \)), the inequality \( |5 - 2x| \leq -3x \) cannot hold because the absolute value is non-negative, while the right-hand side is non-positive.

Given that the right-hand side is negative when \( x \geq 0 \), there's **no solution** in that range because no non-negative value can be less than or equal to a negative number.

Now, let's proceed with the case where \( x < 0 \).

### Case 1: When \( x < 0 \)

Here we solve the two branches of the absolute value inequality:

#### Case 1a: \( 5 - 2x \geq 0 \), which means \( 5 \geq 2x \) or \( x \leq \frac{5}{2} \)
In this case:
\[
5 - 2x \leq -3x
\]
Simplifying:
\[
5 \leq -x \implies x \leq -5
\]

Since we are already assuming \( x < 0 \), this is valid for \( x \leq -5 \).

#### Case 1b: \( 5 - 2x < 0 \), which means \( 5 < 2x \) or \( x > \frac{5}{2} \)
In this case:
\[
-(5 - 2x) \leq -3x \implies -5 + 2x \leq -3x
\]
Simplifying:
\[
2x + 3x \leq 5 \implies 5x \leq 5 \implies x \leq 1
\]

But, since this case applies when \( x > \frac{5}{2} \), there is no solution from this inequality.

### Conclusion:

The only valid solution is \( x \leq -5 \).

### Final Answer:
\[
x \leq -5
\]

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Would you like further details, or do you have any questions about this? Here are five related questions:

1. How do you handle absolute value inequalities in general?
2. What is the difference between strict and non-strict inequalities?
3. How can we graph the solution \( x \leq -5 \)?
4. Can this inequality be solved using a different method?
5. What happens if we have an absolute value inequality with variables on both sides?

**Tip**: When solving absolute value inequalities, always consider the sign of the expression outside the absolute value to determine if the inequality is valid!

Explore how to solve the absolute value inequality |5 - 2x| ≤ -3x with a detailed explanation. This problem covers algebraic concepts including absolute value and linear inequalities, providing insights into their properties. Suitable for students in Grades 9-10.

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