(1) \( x-5=\frac{3}{x} \) (3) \( \frac{3}{x+2}+\frac{4 x-8}{3}=x \) (5) \( \frac{3 x^{2}-9}{x^{2}-4}=4-\frac{2}{x+2} \) (7) \( \frac{x^{2}-3 x-7}{x^{2}-x-2}=\frac{x+2}{x+1}-1 \) (9) \( \frac{x}{2}+\frac{2}{2}= \)

Solving equations algebraically has a rich history dating back to ancient civilizations. The Babylonians had methods to tackle equations as early as 2000 BC, and they used geometric representations to solve quadratic equations. Fast forward to Islamic mathematicians in the 9th century, who made significant advancements in algebra and introduced symbolic notation, paving the way for modern algebra as we know it today! When tackling equations, it’s crucial to double-check your work to avoid simple miscalculations. A common mistake is neglecting to consider restrictions on the variable, especially when dealing with fractions, leading to extraneous solutions. Always look out for divisions by zero, and don’t forget to verify your final answers in the original equation—it's the best way to confirm your solution is indeed correct!