Let's re-examine the inequality and solve it step-by-step to determine the correct set.

Problem Statement

We need to find the solution set of the inequality:

$|6 - 5x| \leq 10$

Step 1: Remove the Absolute Value

Rewrite the absolute value inequality as a compound inequality:

$-10 \leq 6 - 5x \leq 10$

Step 2: Solve the Compound Inequality

Let's split the compound inequality and solve it:

1. Left Side:

   $-10 \leq 6 - 5x$

   Subtract 6 from both sides:

   $-10 - 6 \leq -5x$

   Simplify:

   $-16 \leq -5x$

   Divide by $-5$ (flip the inequality sign):

   $\frac{-16}{-5} \geq x$

   Simplify:

   $\frac{16}{5} \geq x \quad \text{or} \quad x \leq \frac{16}{5}$

2. Right Side:

   $6 - 5x \leq 10$

   Subtract 6 from both sides:

   $-5x \leq 4$

   Divide by $-5$ (flip the inequality sign):

   $x \geq \frac{4}{-5}$

   Simplify:

   $x \geq -\frac{4}{5}$

Step 3: Combine the Results

Combine the two inequalities:

$-\frac{4}{5} \leq x \leq \frac{16}{5}$

Step 4: Match the Solution to the Choices

The correct answer is:

Option c: $-\frac{4}{5} \leq x \leq \frac{16}{5}$

Would you like to see more details or have any other questions?

Related Questions:

1. How do you solve compound inequalities involving absolute values?
2. What are the key steps when dealing with absolute value inequalities?
3. How does flipping the inequality sign affect the solution?
4. Can you explain why absolute values split into two inequalities?
5. How would this process change if the inequality were $|6 - 5x| > 10$?

Tip:

Always check your final answer by substituting boundary values into the original inequality to ensure the solution set is correct!