91Ó°ÊÓ

Set-builder notation is a mathematical shorthand for describing a set by specifying the properties that its elements must satisfy. Instead of listing out individual elements, it uses a variable and a condition to define the members of a set. For instance, to translate an interval into set-builder notation, first identify the boundary numbers and whether they are included or not.

For the interval \[[-5,2)\], it's translated into set-builder notation as \{ x | -5 \leq x < 2 \}. This means we're looking at all values of \(x\) that are greater than or equal to -5 and less than 2. The square bracket around -5 indicates that it's a part of the set (\(x\) is allowed to equal -5), while the parenthesis around 2 indicates that 2 itself is not part of the set (\(x\) must be strictly less than 2).

When writing in set-builder notation, remember to:

• Always use a variable, commonly \(x\), to represent the elements of the set.
• Include a vertical bar \( | \) or colon (\( : \)) which can be read as 'such that.'
• Provide the condition(s) that the variable must satisfy to be included in the set.
• Pay close attention to less than/greater than symbols and whether they are strict (<, >) or inclusive (\leq, \geq).

Graphing an interval on a number line provides a visual representation of the set defined by that interval. To sketch the interval \[[-5,2)\], start by drawing a horizontal line, which will represent your number line, and mark the scale with numbers. Next, identify the critical points of the interval; in this case, the numbers -5 and 2.

On the number line, -5 is included in the interval so you will represent it with a solid dot, or closed circle. Since 2 is not included, it will be represented by an open circle. Then, draw a continuous line between the two points to indicate that every number between -5 and 2 is part of the set, but not including 2. Ensure the following for accuracy:

• Use a closed circle for numbers that are included in the set.
• Use an open circle for numbers that are not included in the set.
• A solid line between these points shows the range of numbers that are part of the set.

This visual method helps you quickly see which numbers are parts of the interval and enhances your understanding of inequalities and set boundaries.

Interpreting intervals correctly is a vital skill in precalculus, as it helps form an understanding of sets, functions, and inequalities. An interval is a way to describe a range of numbers that lies between two endpoints. The notation \[[-5,2)\] is such an interval, with the square bracket indicating that the lower endpoint -5 is included in the set, while the parenthesis shows that the upper endpoint 2 is not.

To fully interpret this interval, consider the following:

• An interval can be one of four types: open (neither endpoint is included), closed (both endpoints are included), or half-open/half-closed (one endpoint is included, and the other is not).
• The interval \[[-5,2)\] is a half-closed interval, where -5 is the lower boundary and included, represented by a closed circle on the graph, and 2 is the upper boundary and excluded, represented by an open circle on the graph.
• This interval represents all real numbers from -5 up to, but not including, 2.

The skill to interpret intervals helps make sense of domain and range of functions, integrals in calculus, and various other concepts in mathematics.