Let's analyze each option to determine which equation has exactly one real solution:

Option A: $2(x + 10) = 2x + 20$

Expand $2(x + 10)$: $2x + 20 = 2x + 20$ Simplify by subtracting $2x + 20$ from both sides: $0 = 0$ This equation is an identity, meaning it is true for all values of $x$ (infinite solutions).

Option B: $7x + 10 = 7x - 19$

Simplify by subtracting $7x$ from both sides: $10 = -19$ This is a contradiction, meaning there is no solution.

Option C: $2x + 10 = 2x + 10$

Simplify by subtracting $2x + 10$ from both sides: $0 = 0$ This equation is also an identity, meaning it is true for all values of $x$ (infinite solutions).

Option D: $7x + 10 = -2x - 19$

Simplify by moving all terms involving $x$ to one side: $7x + 2x = -19 - 10$ $9x = -29$ Solve for $x$: $x = -\frac{29}{9}$ This equation has exactly one real solution.

Final Answer:

The correct option is: D. $7x + 10 = -2x - 19$

Would you like further clarification or to explore related questions?

Here are 5 related questions to expand on this:

1. What are examples of equations with infinite solutions?
2. How can you determine if an equation has no solution without solving it fully?
3. How do equations with exactly one solution differ from those with infinite solutions?
4. What happens if you introduce a quadratic term to one of these equations?
5. Can we determine solutions graphically? How?

Tip: For linear equations, check if terms cancel out or form contradictions to quickly determine the type of solution set.