91Ó°ÊÓ

Interval notation is a way to represent the solutions of inequalities succinctly using brackets and parentheses. It shows the range that a variable, usually denoted as "x", can take. In this method, round brackets \(()\) are used to signify that an endpoint is not included (open interval), while square brackets \([]\) indicate that the endpoint is included (closed interval). In our exercise, the inequality \(|2x| > 7\) results in two separate intervals:

• \(x < -\frac{7}{2}\): This shows that "x" can take any value less than -3.5. The interval is expressed as \((-\infty, -\frac{7}{2})\).
• \(x > \frac{7}{2}\): Similarly, "x" can also take values greater than 3.5, represented as \((\frac{7}{2}, \infty)\).

These two intervals indicate the solution for the inequality \(|2x| > 7\). The union symbol \(\cup\) is used to combine them, representing elements that satisfy either condition, giving us the final solution: \((-\infty, -\frac{7}{2}) \cup (\frac{7}{2}, \infty)\).

Graphing inequalities provides a visual representation of the solution set on a number line. It helps in quickly identifying the ranges of values that satisfy the inequality. For the inequality \(|2x| > 7\), we break it into \(x < -\frac{7}{2}\) and \(x > \frac{7}{2}\), which creates two separate regions on the graph.To graph these inequalities:

• Start by drawing a number line.
• Identify and mark the critical numbers: \(-\frac{7}{2}\) and \(\frac{7}{2}\).
• Since the intervals \((-\infty, -\frac{7}{2})\) and \((\frac{7}{2}, \infty)\) are open, use open circles on the number line at these points to show the values are not included in the solution set.
• Draw an arrow extending to the left from \(-\frac{7}{2}\) to represent all numbers less than \(-\frac{7}{2}\).
• Draw another arrow extending to the right from \(\frac{7}{2}\) to indicate all numbers greater than \(\frac{7}{2}\).

Each arrow signifies the range of solutions, clearly visualizing where "x" can lie on the number line.

The solution set for an inequality consists of all values that make the inequality true. For the absolute value inequality \(|2x| > 7\), the solution set is made up of two distinct parts:

• All values of "x" less than \(-\frac{7}{2}\).
• All values of "x" greater than \(\frac{7}{2}\).

In set representation, these are often represented with inequalities like \(x < -\frac{7}{2}\) or \(x > \frac{7}{2}\). However, interval notation provides a more compact and standardized way to express these. Thus, the solution set is written as \((-\infty, -\frac{7}{2}) \cup (\frac{7}{2}, \infty)\).Using two separate intervals linked with a union symbol (\(\cup\)) shows all possible values of "x" that solve the inequality, highlighting non-continuous ranges of solutions. This clear representation makes evaluating and communicating solutions more systematic and understandable, especially when translating from a mathematical solution to a practical context.