The inequality is:

$\frac{2x - 3}{(x - 2)(x - 4)} \leq 0$

To solve this inequality, we can follow these steps:

Step 1: Find the critical points

The critical points occur when the numerator or denominator is equal to zero. These points will divide the number line into intervals to test.

• Numerator (2x - 3 = 0): $2x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2}$

• Denominator ((x - 2)(x - 4) = 0): $(x - 2)(x - 4) = 0 \quad \Rightarrow \quad x = 2 \quad \text{or} \quad x = 4$

So, the critical points are $x = \frac{3}{2}, 2, 4$.

Step 2: Analyze the intervals

These critical points divide the number line into four intervals:

1. $(-\infty, \frac{3}{2})$
2. $\left(\frac{3}{2}, 2\right)$
3. $(2, 4)$
4. $(4, \infty)$

We will now test the sign of the expression $\frac{2x - 3}{(x - 2)(x - 4)}$ in each interval.

• For $(-\infty, \frac{3}{2})$: Choose $x = 1$ $\frac{2(1) - 3}{(1 - 2)(1 - 4)} = \frac{-1}{(-1)(-3)} = \frac{-1}{3} < 0$

• For $\left(\frac{3}{2}, 2\right)$: Choose $x = \frac{7}{4}$ $\frac{2\left(\frac{7}{4}\right) - 3}{\left(\frac{7}{4} - 2\right)\left(\frac{7}{4} - 4\right)} = \frac{\frac{7}{2} - 3}{\left(\frac{7}{4} - \frac{8}{4}\right)\left(\frac{7}{4} - \frac{16}{4}\right)} = \frac{\frac{1}{2}}{\left(\frac{-1}{4}\right)\left(\frac{-9}{4}\right)} > 0$

• For $(2, 4)$: Choose $x = 3$ $\frac{2(3) - 3}{(3 - 2)(3 - 4)} = \frac{6 - 3}{(1)(-1)} = \frac{3}{-1} = -3 < 0$

• For $(4, \infty)$: Choose $x = 5$ $\frac{2(5) - 3}{(5 - 2)(5 - 4)} = \frac{10 - 3}{(3)(1)} = \frac{7}{3} > 0$

Step 3: Determine inclusion of critical points

The inequality is $\leq 0$, so we need to check when the expression is equal to zero. This happens when $2x - 3 = 0$, i.e., $x = \frac{3}{2}$.

For the points where the denominator is zero ($x = 2$ and $x = 4$), the expression is undefined, so these points cannot be included in the solution.

Step 4: Write the solution

The expression is negative or zero on the intervals $(-\infty, \frac{3}{2}]$ and $(2, 4)$, but excluding the points $x = 2$ and $x = 4$. Therefore, the solution is:

$x \in (-\infty, \frac{3}{2}] \cup (2, 4)$

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Would you like more details on any of the steps? Here are five related questions to deepen understanding:

1. How can you determine whether a rational inequality is less than or greater than zero?
2. What happens to the inequality if the numerator or denominator is negative?
3. How would the solution change if the inequality was $\geq 0$?
4. Can critical points from the denominator ever be included in the solution of a rational inequality?
5. How can you graph the solution set of a rational inequality?

Tip: Always check where the denominator becomes zero when solving rational inequalities to avoid including undefined points in the solution.