https://file.aiquickdraw.com/mathbot/file/1726599429546qy166hlr.jpg

The inequality presented in the image is:

\[
11 - \frac{3x}{2} < \frac{5x + 14}{3} \leq \frac{9}{5}(2 + x)
\]

We will solve this step by step to determine the solution set.

### Step 1: Break the inequality into two parts
We have two parts of the inequality:
1. \( 11 - \frac{3x}{2} < \frac{5x + 14}{3} \)
2. \( \frac{5x + 14}{3} \leq \frac{9}{5}(2 + x) \)

### Step 2: Solve the first inequality \( 11 - \frac{3x}{2} < \frac{5x + 14}{3} \)

Let's start by multiplying both sides by the least common denominator (LCD) of 2 and 3, which is 6.

\[
6 \left( 11 - \frac{3x}{2} \right) < 6 \left( \frac{5x + 14}{3} \right)
\]

This simplifies to:

\[
66 - 9x < 2(5x + 14)
\]

Now expand the right side:

\[
66 - 9x < 10x + 28
\]

Bring all terms involving \(x\) to one side and constants to the other:

\[
66 - 28 < 10x + 9x
\]

\[
38 < 19x
\]

Solve for \(x\):

\[
x > \frac{38}{19} \quad \Rightarrow \quad x > 2
\]

### Step 3: Solve the second inequality \( \frac{5x + 14}{3} \leq \frac{9}{5}(2 + x) \)

Multiply both sides by the least common denominator, which is 15:

\[
15 \left( \frac{5x + 14}{3} \right) \leq 15 \left( \frac{9}{5}(2 + x) \right)
\]

This simplifies to:

\[
5(5x + 14) \leq 3(9)(2 + x)
\]

Expand both sides:

\[
25x + 70 \leq 54 + 27x
\]

Now bring all terms involving \(x\) to one side and constants to the other:

\[
70 - 54 \leq 27x - 25x
\]

\[
16 \leq 2x
\]

Solve for \(x\):

\[
x \geq \frac{16}{2} \quad \Rightarrow \quad x \geq 8
\]

### Step 4: Combine the results
From the first inequality, we found \(x > 2\). From the second inequality, we found \(x \geq 8\).

Thus, the solution is:

\[
x \geq 8
\]

The solution set is \( [8, \infty) \).

Would you like more details, or do you have any other questions?

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Here are 5 related questions for further practice:
1. How do you solve compound inequalities involving absolute values?
2. What happens when you multiply or divide an inequality by a negative number?
3. How do you graph the solution set of an inequality on a number line?
4. How can inequalities be used to solve word problems in real-life situations?
5. What are the steps to solving inequalities with quadratic terms?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing both sides by a negative number.

Determine el conjunto solución de la siguiente inecuación: 11 - (3x/2) < (5x + 14)/3 ≤ (9/5)(2 + x)

Learn how to solve the compound inequality 11 - (3x/2) < (5x + 14)/3 ≤ (9/5)(2 + x) with a detailed step-by-step solution. This problem covers key algebraic concepts such as inequalities, linear equations, and properties of inequalities. Suitable for students in Grades 9-12.

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