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Interval notation is a method of writing down a range of numbers that are typically solutions to inequalities. In interval notation, we use brackets and parentheses to describe the set of numbers that are included in the solution. A bracket, like \[ or \], indicates that the number is included in the interval, also referred to as a closed interval. A parenthesis, like \((\) or \()\), signifies that the number is not included, making it an open interval.

For example, when we express the solution \(-3 \leq x < \frac{5}{2}\), the appropriate interval notation is \([-3, \frac{5}{2})\). The square bracket on \(-3\) signifies that \(-3\) is included in the set, because of the \(\leq\) symbol. The parenthesis on \(\frac{5}{2}\) shows that while values can approach \(\frac{5}{2}\), they can never actually be \(\frac{5}{2}\), indicated by the \(<\) symbol. Interval notation is an efficient way to communicate the set of possible values for a variable and is widely used in mathematics.

Inequality solutions involve finding the range of values that a variable can take to satisfy a given inequality. Solving inequalities often requires a similar set of steps to solving equations, but with an additional consideration for the direction of the inequality. For instance, when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality flips.

To correctly solve the inequality \(0 < 5-2x \leq 11\), we first isolate \(x\) to find individual constraints \(x < \frac{5}{2}\) and \(x \geq -3\). We then combine these constraints to find the intersection, which represents the set of all values that satisfy both parts of the compound inequality. In this exercise, the solution to \(x\) lies in the range where it is both larger than or equal to -3 and smaller than \(\frac{5}{2}\).

Algebraic manipulation is a skill fundamental to solving inequalities, which involves rearranging algebraic expressions and equations to isolate the variable we're solving for. Crucial steps in manipulation include adding or subtracting the same quantity from both sides of the inequality, as well as multiplying or dividing both sides by the same non-zero number.

In our example, to get from \(2x < 5\) to \(x < \frac{5}{2}\), we divide both sides by 2. Similarly, we take \(-2x \leq 6\) and divide by -2 to get \(x \geq -3\), remembering to flip the inequality sign because of the division by a negative. Mastery of algebraic manipulation is essential to progress in mathematics, as it is used not only in solving inequalities but in simplifying expressions and solving equations as well.