We are given the inequality:

$\frac{4}{5 - x} \leq \frac{5}{2 - x}$

Step 1: Find the values where the denominators are undefined.

The denominators $5 - x$ and $2 - x$ must not be zero, so:

1. $5 - x = 0$ gives $x = 5$
2. $2 - x = 0$ gives $x = 2$

Therefore, the inequality is undefined at $x = 5$ and $x = 2$.

Step 2: Eliminate the denominators

To clear the fractions, multiply both sides of the inequality by $(5 - x)(2 - x)$. However, this is only valid where $(5 - x)(2 - x) \neq 0$, so $x \neq 5$ and $x \neq 2$.

Multiplying both sides:

$4(2 - x) \leq 5(5 - x)$

This simplifies to:

$8 - 4x \leq 25 - 5x$

Step 3: Solve the inequality

Now, solve for $x$:

$8 - 4x \leq 25 - 5x$

First, add $5x$ to both sides:

$8 + x \leq 25$

Next, subtract 8 from both sides:

$x \leq 17$

Step 4: Combine the restrictions

We now know that $x \leq 17$, but we also have the restrictions that $x \neq 5$ and $x \neq 2$, because the original expression is undefined at those points.

Step 5: Solution in interval notation

The solution set is all values of $x$ such that $x \leq 17$, excluding $x = 5$ and $x = 2$. In interval notation, this is:

$(-\infty, 2) \cup (2, 5) \cup (5, 17]$

This is the solution to the inequality.

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Would you like more details on any step or further clarifications?

Here are some related questions:

1. How do we handle inequalities when dividing or multiplying by negative numbers?
2. What happens if the inequality sign changes when solving similar inequalities?
3. Can inequalities have more than two intervals in their solution?
4. How can you visualize the solution of an inequality on a number line?
5. What are some common pitfalls when solving rational inequalities?

Tip: Always check for points where the function is undefined when solving inequalities involving rational expressions!