https://file.aiquickdraw.com/mathbot/file/1726599227322hrlum5z1.jpg

The inequality given in the image is:

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

We need to solve this compound inequality and express the solution in interval notation. Let's break it down step by step.

### Step 1: Solve the left-hand inequality

\[
1 - \frac{3x}{2} \leq \frac{5x + 14}{3}
\]

To solve this, first eliminate the fractions by multiplying both sides by 6 (the least common multiple of 2 and 3):

\[
6 \left(1 - \frac{3x}{2}\right) \leq 6 \left(\frac{5x + 14}{3}\right)
\]

This simplifies to:

\[
6 - 9x \leq 2(5x + 14)
\]

Expanding the terms:

\[
6 - 9x \leq 10x + 28
\]

Now, move all terms involving \(x\) to one side and constant terms to the other:

\[
6 - 28 \leq 10x + 9x
\]

Simplify:

\[
-22 \leq 19x
\]

Now divide by 19:

\[
x \geq -\frac{22}{19}
\]

### Step 2: Solve the right-hand inequality

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

First, eliminate the fractions by multiplying both sides by 15 (the least common multiple of 3 and 5):

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

This simplifies to:

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

Expand both sides:

\[
25x + 70 \leq 54(2 + x)
\]

Now distribute the 54:

\[
25x + 70 \leq 108 + 54x
\]

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

\[
70 - 108 \leq 54x - 25x
\]

Simplify:

\[
-38 \leq 29x
\]

Now divide by 29:

\[
x \geq -\frac{38}{29}
\]

### Step 3: Combine the solutions

From both parts of the compound inequality, we have:

\[
x \geq -\frac{22}{19} \quad \text{and} \quad x \leq -\frac{38}{29}
\]

However, we must carefully check whether these intervals overlap or if further simplification is needed.

I'll finalize the calculation now.The two fractions are:

\[
x \geq -\frac{22}{19} \quad \text{and} \quad x \leq -\frac{38}{29}
\]

Since \(-\frac{22}{19} \approx -1.1579\) and \(-\frac{38}{29} \approx -1.3103\), we can see that \(-\frac{22}{19}\) is greater than \(-\frac{38}{29}\).

Thus, there is no overlap, and no valid solution exists for \(x\). Therefore, the solution set is empty.

The answer in interval notation is:

\[
\emptyset
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you solve compound inequalities with fractions?
2. What does it mean for an interval to be empty?
3. How can we eliminate fractions in inequalities?
4. What are common pitfalls when solving compound inequalities?
5. How do we interpret inequalities graphically?

**Tip:** Always check for overlaps between intervals when solving compound inequalities to ensure the solution is valid.

Determine el conjunto solución de la siguiente inecuación: 1 - (3x/2) <= (5x + 14)/3 <= (9/5)(2 + x)

Learn how to solve the compound inequality 1 - (3x/2) <= (5x + 14)/3 <= (9/5)(2 + x). The solution involves simplifying the inequality step by step, eliminating fractions, and interpreting the solution in interval notation. This problem is suitable for students in Grades 10-12.

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