Let's solve the equation step by step to determine whether it has infinitely many solutions, no solution, or exactly one solution.

The equation is:

$4(5x - 7) = -4(-5x + 7)$

Step 1: Distribute both sides

On the left-hand side:

$4(5x - 7) = 4 \cdot 5x - 4 \cdot 7 = 20x - 28$

On the right-hand side:

$-4(-5x + 7) = -4 \cdot (-5x) - 4 \cdot 7 = 20x - 28$

So, the equation simplifies to:

$20x - 28 = 20x - 28$

Step 2: Subtract $20x$ from both sides

$20x - 28 - 20x = 20x - 28 - 20x$ $-28 = -28$

Conclusion:

This results in a true statement, $-28 = -28$, which holds for any value of $x$.

Thus, the equation has infinitely many solutions.

---

Would you like any further details or have questions about this process?

Here are 5 related questions for further exploration:

1. What is an example of an equation with no solution?
2. How do you identify when an equation has exactly one solution?
3. What happens if we subtract different terms from both sides of an equation?
4. How do you solve systems of linear equations with infinitely many solutions?
5. What is the geometric interpretation of infinitely many solutions?

Tip: When solving equations, if both sides simplify to the same constant (like $-28 = -28$), it indicates infinitely many solutions.