To solve question 18, let's go through it step-by-step. The question asks us to find the possible values of $x$ for the inequality:

$\left| \frac{x}{2} + 4 \right| < \frac{2x + 4}{3}$

1. Analyze the absolute value expression: Since we have an absolute value, we need to consider two cases for the inequality, one for the positive scenario and one for the negative scenario.

2. Case 1: $\frac{x}{2} + 4 \geq 0$

   • Simplify the inequality without the absolute value.

3. Case 2: $\frac{x}{2} + 4 < 0$

   • Simplify with the negation of the expression inside the absolute value.

Let me go through the calculations for this and I’ll update with the solution.It appears that my initial approach encountered an issue. Let’s break down the inequality manually to ensure clarity:

The inequality we have is:

$\left| \frac{x}{2} + 4 \right| < \frac{2x + 4}{3}$

Step-by-Step Solution:

1. Case 1: $\frac{x}{2} + 4 \geq 0$ (implying $x \geq -8$)

   • In this case, the absolute value expression becomes positive, so we can write: $\frac{x}{2} + 4 < \frac{2x + 4}{3}$
   • Clearing the fractions by multiplying through by 6 gives: $3x + 24 < 4x + 8$
   • Simplifying, we get: $x > 16$

2. Case 2: $\frac{x}{2} + 4 < 0$ (implying $x < -8$)

   • In this case, we need to flip the sign inside the absolute value: $-\left( \frac{x}{2} + 4 \right) < \frac{2x + 4}{3}$
   • This simplifies to: $-\frac{x}{2} - 4 < \frac{2x + 4}{3}$
   • Clearing fractions by multiplying through by 6: $-3x - 24 < 4x + 8$
   • Rearranging terms, we get: $-7x < 32$ $x > -\frac{32}{7} \approx -4.57$

Solution Set

Combining the results from both cases, the solution set for $x$ is:

$x \in \left(-\frac{32}{7}, -8\right) \cup (16, \infty)$

Would you like further clarification on this solution?

Here are some related questions to explore:

1. How can we handle inequalities involving absolute values in general?
2. What other methods are available for solving inequalities with absolute values?
3. How would the solution change if the inequality sign were reversed?
4. Can we graph these solutions on a number line for better visualization?
5. What are some common mistakes to avoid when working with absolute value inequalities?

Tip: Always remember to consider both cases when working with absolute values, as they represent both positive and negative scenarios of the expression inside.