To solve the inequality:

$-5 \leq \frac{5 - 3x}{2} \leq 8$

we can break it down into two separate inequalities:

1. $-5 \leq \frac{5 - 3x}{2}$
2. $\frac{5 - 3x}{2} \leq 8$

Step 1: Solve the first inequality

$-5 \leq \frac{5 - 3x}{2}$

Multiply both sides by 2 to eliminate the denominator: $-10 \leq 5 - 3x$

Now, subtract 5 from both sides: $-15 \leq -3x$

Divide by -3, and remember to reverse the inequality sign because we are dividing by a negative number: $5 \geq x$ or $x \leq 5$

Step 2: Solve the second inequality

$\frac{5 - 3x}{2} \leq 8$

Multiply both sides by 2: $5 - 3x \leq 16$

Subtract 5 from both sides: $-3x \leq 11$

Divide by -3, and reverse the inequality: $x \geq -\frac{11}{3}$

Final Solution

Now we combine the two inequalities: $-\frac{11}{3} \leq x \leq 5$

This means the solution is: $x \in \left[ -\frac{11}{3}, 5 \right]$

Do you want more details or have any questions?

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Here are some related questions to explore:

1. How do you solve compound inequalities with multiple variables?
2. What happens when you divide or multiply inequalities by negative numbers?
3. How would the solution change if the inequality was strict (using "<" and ">")?
4. What are the graphical representations of compound inequalities?
5. Can compound inequalities have no solution?

Tip: Always remember to flip the inequality sign when dividing or multiplying by a negative number in inequalities!