(Secre for Question 2: _of 5 points) 2. Solve the inequality \( -\frac{5}{2}(3 x+4)<6-3 x \). Show your work.

To solve the inequality \( -\frac{5}{2}(3x + 4) < 6 - 3x \), let's start by distributing on the left side. 1. Distribute \( -\frac{5}{2} \): \[ -\frac{5}{2} \cdot 3x - \frac{5}{2} \cdot 4 < 6 - 3x \] This simplifies to: \[ -\frac{15}{2}x - 10 < 6 - 3x \] 2. Next, let's isolate \( x \). First, we'll move \( -3x \) to the left side and add \( 10 \) to both sides: \[ -\frac{15}{2}x + 3x < 6 + 10 \] Simplifying the right side gives us: \[ -\frac{15}{2}x + 3x < 16 \] Rewrite \( 3x \) as \( \frac{6}{2}x \) to combine like terms: \[ -\frac{15}{2}x + \frac{6}{2}x < 16 \] This simplifies to: \[ -\frac{9}{2}x < 16 \] 3. Now, divide both sides by \( -\frac{9}{2} \). Remember, when dividing by a negative number, the inequality sign flips: \[ x > \frac{16 \cdot 2}{9} \] This simplifies to: \[ x > \frac{32}{9} \] So, the solution to the inequality is: \[ x > \frac{32}{9} \] Now, onto some fun facts! In the early days of algebra, inequalities were not formally represented. Mathematicians like René Descartes and Nicolas Bourbaki helped formalize inequality notation. It wasn't until the 20th century that they gained the rigorous treatment we use today, complete with properties and theorems! In the real world, inequalities are everywhere! From economics—like comparing profits and costs—to planning a budget where you want expenses to stay below a certain threshold, inequalities help us understand constraints and make informed decisions. They can guide everything from business strategies to personal finance management!