To determine when the equation $5 + 2 = 5 + x$ has no solution, we first solve it normally.

1. Simplify the left-hand side: $7 = 5 + x$

2. Subtract 5 from both sides: $7 - 5 = x$ $x = 2$

So, if $x = 2$, the equation is valid and does have a solution. However, for the equation to have no solution, this situation must be impossible. But notice that this type of linear equation will always have a solution unless we introduce some inconsistency.

The only way this equation can have no solution is if we assume the structure of the equation is inconsistent by design. For example, consider:

• If you modify the equation to include something that leads to a contradiction, like comparing two constants that are inherently different (e.g., $5 + 2 = 5 + 4$), then you would have no solution because $7 \neq 9$.

Thus, if the equation is set up in such a way that the left and right sides are fundamentally different, like $5 + 2 \neq 5 + x$ for any $x$, then there would be no solution. This happens when the values on both sides are fixed and different.

Would you like to explore this further or have additional questions?

Here are some related questions:

1. What types of equations have no solutions?
2. Can equations with variables in both sides always be solved?
3. How does introducing contradictions affect the solutions of equations?
4. What are the conditions for an equation to have infinitely many solutions?
5. Can inequalities also have no solutions? How?

Tip: When solving for variables, always simplify both sides of the equation first to see if contradictions arise.