We are tasked with solving the inequality:

$5(3 - x) > -2x + 6$

Let's go through the steps.

Step 1: Distribute the $5$ on the left-hand side.

$5 \cdot 3 - 5 \cdot x > -2x + 6$ $15 - 5x > -2x + 6$

Step 2: Move all the $x$-terms to one side and constants to the other.

Add $5x$ to both sides: $15 > 3x + 6$

Step 3: Subtract $6$ from both sides:

$9 > 3x$

Step 4: Solve for $x$ by dividing both sides by 3:

$x < 3$

Thus, the solution is: $x < 3$

Final Answer:

$x < 3$

Would you like more details or have any questions about this solution?

Here are 5 related questions to expand your understanding:

1. What happens if the inequality symbol is reversed?
2. How would the solution change if we had a negative coefficient in front of $x$?
3. How would you solve this inequality graphically?
4. Can you try solving a similar inequality like $4(2 - x) \geq -x + 8$?
5. What if the inequality was $5(3 - x) \leq -2x + 6$?

Tip: When dividing or multiplying both sides of an inequality by a negative number, remember to reverse the inequality symbol!