e. \( \frac{5}{10} x+1=\frac{4}{3}-\left(x-\frac{7}{3}\right) \)

First, let's simplify the equation step by step! Start by distributing on the right side: \[ \frac{5}{10} x + 1 = \frac{4}{3} - x + \frac{7}{3} \] Now, combine like terms on the right: \[ \frac{5}{10} x + 1 = \frac{4 + 7}{3} - x = \frac{11}{3} - x \] Next, let's get all terms involving \(x\) on one side and constants on the other. Rearranging gives: \[ \frac{5}{10} x + x = \frac{11}{3} - 1 \] We can express \(\frac{5}{10} x\) as \(\frac{1}{2} x\), and \(x\) becomes \(\frac{2}{2} x\), thus: \[ \frac{1}{2} x + \frac{2}{2} x = \frac{11}{3} - \frac{3}{3} \] Combine the left side: \[ \frac{3}{2} x = \frac{8}{3} \] To isolate \(x\), multiply both sides by \(\frac{2}{3}\): \[ x = \frac{8}{3} \cdot \frac{2}{3} = \frac{16}{9} \] So, the solution is \( x = \frac{16}{9} \). --- Did you know that fractions have been around since ancient times? The Egyptians and Babylonians were using fractions as far back as 3000 BC! In fact, they had their own unique ways of representing these curious little numbers, making math a fascinating journey through history. In the real world, knowing how to manipulate fractions is super useful! Whether you’re cooking and need to adjust a recipe, splitting a bill among friends, or calculating interest rates, these skills come in handy! So, next time you're in the kitchen or budgeting, remember your fraction-fighting skills and unleash your inner math wizard!