Solving inequalities with no solution or all real numbers as solutions For each inequality, choose the statement that describes its solution. If applicable, give the solution. \( \begin{array}{l}\text { (a) }-2(w+6)+19>2(5-w) \\ \text { No solution } \\ \text { All real numbers are solutions }\end{array} \)

To solve the inequality \(-2(w+6)+19>2(5-w)\), let's simplify both sides. Expanding gives \(-2w - 12 + 19 > 10 - 2w\), which simplifies to \(-2w + 7 > 10 - 2w\). When we isolate the terms, we see that the \(w\) terms cancel out, leading to \(7 > 10\), a statement that is never true. Hence, there is no solution. Now, if you encounter an inequality that simplifies to an identity, like \(2x + 3 = 2x + 3\), this means all real numbers satisfy the inequality. In such cases, it's simply a matter of recognizing that the variables cancel out, leaving a true statement that stands for all values. This is the beauty of inequalities!