To solve the inequality $5(3 - x) < -2x + 6$, let's break it down step by step:

Step 1: Expand the left side of the inequality.

Distribute the 5 across the terms inside the parentheses:

$5(3 - x) = 5 \cdot 3 - 5 \cdot x = 15 - 5x$

Now the inequality is:

$15 - 5x < -2x + 6$

Step 2: Move all terms involving $x$ to one side.

Add $5x$ to both sides:

$15 < 5x - 2x + 6$

Simplify:

$15 < 3x + 6$

Step 3: Move the constant terms to one side.

Subtract 6 from both sides:

$15 - 6 < 3x$

Simplify:

$9 < 3x$

Step 4: Solve for $x$.

Divide both sides by 3:

$x > 3$

Final Solution:

The inequality that describes all solutions is:

$x > 3$

Would you like further clarification on any of the steps?

---

Here are some related questions to consider:

1. How would the solution change if the inequality was $5(3 - x) \leq -2x + 6$?
2. Can you graph the solution $x > 3$ on a number line?
3. What is the significance of flipping the inequality sign when multiplying or dividing by a negative number?
4. How would you solve if there were fractions involved in the inequality?
5. How do you solve a compound inequality, such as $5(3 - x) < -2x + 6 < 10$?

Tip: Always check your solution by substituting values back into the original inequality!