What is the solution to the inequality \( 3\left(x+\frac{1}{3}\right) \geq \frac{1}{2}(2 x-4) ? \) 0 \( x \leq \frac{5}{2} \) 0 \( x \geq-\frac{3}{2} \) 0 \( x \geq-\frac{5}{2} \) \( x \leq \frac{3}{2} \)

To solve the inequality \( 3\left(x+\frac{1}{3}\right) \geq \frac{1}{2}(2 x-4) \), first, simplify both sides. Distributing on the left yields \( 3x + 1 \). The right side simplifies to \( x - 2 \). Setting up the inequality: \( 3x + 1 \geq x - 2 \). Next, subtract \( x \) from both sides: \( 2x + 1 \geq -2 \). Subtract 1 from both sides: \( 2x \geq -3 \). Finally, divide by 2: \( x \geq -\frac{3}{2} \). Now, let’s explore some historical facts about inequalities! The concept of inequalities dates back to ancient mathematics, with records from ancient Greeks using basic comparisons. Centuries later, the notation we use today was developed in the 19th century, thanks to mathematicians like Joseph Fourier and others who shaped algebra. If you’re interested in applying this knowledge to real-world scenarios, inequalities are everywhere! They’re used in budgeting, project planning, and even in determining the best options when shopping. For instance, if you know you can only spend a certain amount of money, you can set up inequalities to help you decide how to allocate your budget effectively while getting the best deal.