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Interval notation is a system used to describe the set of numbers that satisfies an equation or inequality. It provides a concise way to show all the numbers between two endpoints. For the inequality in our exercise, the solution is expressed as \( x \leq \frac{3}{2} \), which means the set of all numbers that are less than or equal to \( \frac{3}{2} \). In interval notation, this is written as \( (-\infty, \frac{3}{2}] \). The parentheses around \( -\infty \) indicate that negative infinity is not a number, but rather a concept that describes values getting indefinitely smaller. The square bracket next to \( \frac{3}{2} \) signifies that \( \frac{3}{2} \) is included in the set. Therefore, the interval \( (-\infty, \frac{3}{2}] \) represents all real numbers up to and including \( \frac{3}{2} \).

When it comes to other types of inequalities, open intervals (denoted by parentheses) and closed intervals (denoted by square brackets) help distinguish between values that are not included in the solution set (open) and values that are (closed). Understanding interval notation is an essential part of precalculus and aids in clear communication of mathematical concepts.

Inequality manipulation involves using algebraic operations to find the solution set of an inequality. This process is similar to solving equations, with a key consideration being the direction of the inequality. Let's break down the steps from our exercise.

In Step 1, subtracting 4 from both sides did not affect the direction of the inequality since subtraction is a 'safe' operation—that is, it maintains the order. The operation resulted in \(2x \leq 3\). In Step 2, dividing both sides by a positive number, in this case, 2, also maintained the inequality's direction, giving us \(x \leq \frac{3}{2}\). It's crucial to note that if we were to multiply or divide by a negative number, we would reverse the inequality's direction. It's this careful manipulation of inequalities that allows us to isolate the variable and find the valid range of values that satisfy the original inequality.

Precalculus serves as the foundation that bridges the gap between algebraic concepts and the more advanced studies of calculus. In our exercise, we've encountered a precalculus topic involving inequalities and expressing solutions in interval notation. These concepts mandate a solid understanding of algebraic principles, such as performing operations on both sides of an inequality and knowing when these operations affect the inequality's direction.

Precalculus also emphasizes the importance of function behavior, the analysis of graphs, and the nature of sequences and series. It prepares students for the calculative rigor they will encounter in calculus by reinforcing core mathematical skills and introducing more advanced topics like limits and the beginnings of differential and integral calculus. Grasping these precalculus concepts is crucial for success in higher mathematics and for developing a deep understanding of how mathematics describes the world around us.