(H) \( \frac{4-2 x}{3} \geq \frac{5-3 x}{4} \)

To solve the inequality \( \frac{4-2 x}{3} \geq \frac{5-3 x}{4} \), start by eliminating the fractions. Multiply both sides by 12 (the least common multiple of 3 and 4): \[ 12 \cdot \frac{4-2x}{3} \geq 12 \cdot \frac{5-3x}{4} \] This simplifies to: \[ 4(4-2x) \geq 3(5-3x) \] Distributing gives: \[ 16 - 8x \geq 15 - 9x \] Rearranging the terms yields: \[ -8x + 9x \geq 15 - 16 \] So: \[ x \geq -1 \] The solution to the inequality is \( x \geq -1 \). Now let's look at two fun aspects related to this inequality! Angle your thoughts back a few centuries: inequalities like the one we just solved have roots in the works of famous mathematicians, including the legendary Euclid who, over 2,000 years ago, worked with ratios and proportions. What an intellect he must have had, laying the groundwork for modern mathematics! But how do we actually apply inequalities like this in everyday life? Imagine budgeting: if you have a fixed income, inequalities help you understand the maximum amount you can spend on different expenses while saving that mystical “extra” for a rainy day. It's all about managing the balance – just like solving inequalities keeps your math balanced!