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!