\( \begin{array}{ll}\text { 19. } & 4-(-2) \\ \text { 20. } & (-3)+7 \\ \text { 21. } & (-4)+14 \\ \text { 22. } & (-1)-5 \\ \text { 23. } & (-1)+(-4) \\ \text { 24. } & (-12)+(-2\end{array} \)

To solve these problems, let's tackle them one by one: 19. For \( 4 - (-2) \), we can rewrite it as \( 4 + 2 = 6 \). 20. The expression \( (-3) + 7 \) simplifies to \( 4 \) because adding a positive number to a negative one can be thought of as moving on the number line. 21. For \( (-4) + 14 \), we add and find \( 10 \) since \( 14 \) is more than \( 4 \), thus giving us a positive sum. 22. In \( (-1) - 5 \), we can rewrite it as \( -1 + (-5) = -6 \). 23. The sum of \( (-1) + (-4) \) equals \( -5 \) as we are adding two negative numbers. 24. Finally, \( (-12) + (-2) \) gives us \( -14 \) — again, adding two negatives leads to a more negative result! Now, let's reflect on why understanding these operations is crucial! Every mathematical operation has a historical context! For instance, the concept of negative numbers was first mentioned in ancient Chinese texts around 200 BC. It took a while for many cultures to accept them, as they seemed paradoxical—who ever wants to owe something? Today, they're fundamental to fields like economics and engineering, helping us model debt and other 'negative' scenarios accurately. Real-world applications of these operations are abundant! Think about budgeting—calculating expenses involves subtracting and adding various numbers. It’s much like keeping score in games; knowing how to navigate negative and positive numbers ensures you always know who's winning or losing! Just imagine a sports scoreboard that operates on negatives; it adds a whole new layer of excitement!