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Interval notation is a way of writing subsets of the real number line. It is often used to represent solution sets for inequalities. Since interval notation succinctly shows the beginning and end of a set, it's both efficient and clear.
When writing in interval notation, we use brackets depending on whether the endpoint is included:

• Square brackets "[" or "]" indicate that the endpoint is included in the set.
• Parentheses "(" or ")" indicate that the endpoint is not included in the set.

When we solved the inequality \([-\frac{17}{10}, -\frac{13}{10}]\), both numbers are included in the solution set, thus square brackets are used. This tells us that our variable, \(x\), includes all values from \(-\frac{17}{10}\) to \(-\frac{13}{10}\).
Interval notation helps make the range of solutions clear at a glance, making it easier for you to understand which numbers fit an inequality.

Solving inequalities is similar to solving equations but with a few additional rules that need to be followed. For instance, when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.
A step-by-step approach is often helpful:

• Isolate the variable you're solving for.
• Perform algebraic operations step by step.
• Be mindful of reversing the sign when necessary.

In the example inequality \(2\), after isolating \(|2x+3|\), we derived simpler inequalities by breaking down the absolute value expression into two separate, linear inequalities. This approach simplifies the process and prevents errors.

Inequalities involving absolute values require special attention because they express distance from zero. For an inequality like \(|2x+3| \leq \frac{2}{5}\), the expression indicates that \(2x+3\) lies within \(\frac{2}{5}\) units from zero. This means the actual expressions to solve are \(-\frac{2}{5} \leq 2x+3 \leq \frac{2}{5}\).
These are then broken down into two inequalities to solve separately:

• \(2x+3 \leq \frac{2}{5}\)
• \(-\frac{2}{5} \leq 2x+3\)

By solving these, you find the range of solutions for \(x\). The absolute value essentially doubles the work, but once you grasp the core concept, this method is logical and straightforward.

Algebraic expressions are combinations of variables, numbers, and operations. Understanding them is crucial for manipulating equations and inequalities. In this problem, \(|2x+3|\) is an algebraic expression that includes an absolute value.
When working with expressions:

• Simplify wherever possible to make calculations easier.
• Perform operations in a systematic manner to ensure accuracy.

For example, isolating the algebraic expression \(|2x+3|\) was a critical step in simplifying the inequality and eventually led us to the absolute value inequalities. Mastery of algebraic manipulation also means gaining a solid understanding of how to work with absolute values and inequalities in tandem.