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Set-builder notation is a powerful way to describe a set by specifying a property that its members must satisfy. It's a mathematical shorthand that provides a concise way of expressing sets, especially in mathematics and higher-level algebra. In set-builder notation, the solution to an inequality is written in a form that explicitly describes the conditions for a variable to be a part of the set.

For the inequality problem we solved, we ended with the inequality: \( x < \frac{21}{2} \). In set-builder notation, this is expressed as:

• \( \{ x \mid x < \frac{21}{2} \} \)

This notation reads as "the set of all \( x \) such that \( x \) is less than \( \frac{21}{2} \)."

The vertical bar "\(|\)" is used to mean "such that," making it clear what properties the elements of the set must satisfy. This clarity is one reason set-builder notation is favored when describing complex sets or solutions.

Interval notation is an efficient method of denoting a range of values on the number line. It uses simple symbols to express a set of numbers that satisfy an inequality. In our solved exercise, we end with the inequality \( x < \frac{21}{2} \).

In interval notation, this is written as:

• \(( -\infty, \frac{21}{2} )\)

The rounded parenthesis \(()\) indicate that \( -\infty \) and \( \frac{21}{2} \) are not included in the set. This contrasts with square brackets \([]\), which would mean inclusion.

By using interval notation, mathematicians and students alike can quickly convey the solution to inequalities in a compact form. This method simplifies reading and interpreting mathematical information, especially when dealing with continuous sets of numbers.

Variable isolation is a crucial step in solving equations and inequalities. It involves manipulating an equation or inequality so that the variable is alone on one side of the equation. This gives a clear expression for the solution. In our exercise, we started with the inequality: \[-2x + 21 > 0\].
The process of variable isolation includes several steps:

• First, constants are moved to the other side of the inequality by performing inverse operations. For example, subtracting 21 from both sides results in \(-2x > -21\).
• Next, to isolate \( x \), we divided by \(-2\), remembering to reverse the inequality to \( x < \frac{21}{2} \).

When dividing or multiplying both sides of an inequality by a negative number, it's essential to reverse the inequality sign. This rule helps maintain the inequality's direction and accuracy.

Successfully isolating variables is a foundational skill in algebra that simplifies complex equations and inequalities, allowing students to find solutions more intuitively and accurately.