Let's solve the inequality \( \frac{2(x - 4)}{x} < -4 \).

### Step 1: Simplify the inequality
We begin by distributing the \( 2 \) in the numerator:

\[
\frac{2(x - 4)}{x} = \frac{2x - 8}{x}
\]

Now the inequality becomes:

\[
\frac{2x - 8}{x} < -4
\]

### Step 2: Separate the fraction
We can split the fraction as follows:

\[
\frac{2x - 8}{x} = \frac{2x}{x} - \frac{8}{x} = 2 - \frac{8}{x}
\]

Thus, the inequality becomes:

\[
2 - \frac{8}{x} < -4
\]

### Step 3: Isolate the fraction
Subtract 2 from both sides:

\[
-\frac{8}{x} < -6
\]

### Step 4: Multiply by -1 and reverse the inequality
Multiplying both sides by -1 reverses the inequality:

\[
\frac{8}{x} > 6
\]

### Step 5: Solve the inequality
Now solve for \( x \) by multiplying both sides by \( x \), but keep in mind that the sign of \( x \) determines how the inequality behaves:

- If \( x > 0 \), multiply both sides by \( x \):

\[
8 > 6x \quad \Rightarrow \quad x < \frac{4}{3}
\]

- If \( x < 0 \), multiply both sides by \( x \) (and reverse the inequality again):

\[
8 < 6x \quad \Rightarrow \quad x > \frac{4}{3}
\]

Thus, the solution is \( 0 < x < \frac{4}{3} \).

### Final Solution:
\[
x \in (0, \frac{4}{3})
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What happens if \( x \) is negative in a similar inequality?
2. How would the inequality change if the right-hand side were \( +4 \) instead of \( -4 \)?
3. How do you solve compound inequalities involving fractions?
4. Can you solve the same inequality using a graphical method?
5. What if the numerator was a quadratic expression?

**Tip:** Always be cautious of reversing the inequality sign when multiplying or dividing by negative numbers!

Explore how to solve the inequality 2(x - 4)/x < -4 step-by-step. This solution involves simplifying rational expressions and applying inequality rules. Ideal for students in Grades 9-11 studying Algebra and Inequalities.

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