when we say that a solution is true for $-\infty < x < +\infty$ does that mean that it is true for all real numbers or any number between negative infinity and positive infinity, which is quite vague.
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When we say solutions exist for $-\infty \lt x \lt +\infty$, we are saying the solution is valid for all $x$ in that range. So, your first statement is the correct interpretation.