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Interval notation is a helpful way to express a range of numbers, like solutions to inequalities, using brackets and parentheses. This symbolic form provides a clear and compact representation of the solution set. When we encounter an inequality like \(3 \leq 2x < 7\), solving it leads to an expression of \(1.5 \leq x < 3.5\). The interval notation that represents this solution is \([1.5, 3.5)\).

Here’s a simple guide to understanding interval notation:

• A square bracket \([\) or \(]\) signifies that the endpoint is included in the set. Hence, \([1.5\) means that 1.5 is a part of the solution.
• A parenthesis \((\) or \()\) indicates that the endpoint is not included. Therefore, \()3.5)\) implies that 3.5 is not part of the solution.

It's important to note that "including" means the endpoint is a possible value, while "excluding" means it is not. By using interval notation, you can quickly communicate the span of numbers representing solutions to an inequality.

Graphing an inequality on a number line is an excellent visual method to understand the solution set. This technique helps illuminate the range of possible values that form part of the solution. For the inequality \(1.5 \leq x < 3.5\), we can translate this to a number line.

Here's the process of number line graphing:

• Draw a horizontal line, which will represent the number line.
• Mark points for key values, in this case, 1.5 and 3.5, on the line.
• Since 1.5 is included, denote this with a closed circle at 1.5. Closed circles mean that the value itself is part of the solution.
• Place an open circle at 3.5 since this endpoint is not included in the solution. Open circles highlight which values are not part of the solution.
• Draw a solid line connecting these circles to represent the range of the solution.

Using number line graphing provides a clear visual path of understanding which points meet the criteria of the inequality.

Displaying inequalities through various notations is crucial in understanding and conveying mathematical relationships. An inequality describes a range of values where one side of the inequality is greater or lesser than the other. For example, \(3 \leq 2x < 7\) denotes that \(2x\) must be between 3 and 7.

Here's how to represent inequalities and what they signify:

• The inequality \(\leq\) means "less than or equal to." If you see \( \leq \), it implies that the value can be equal to or greater than the specified number.
• On the flip side, \(<\) represents "less than," indicating that the value must be strictly less than the specified number.

When solving inequalities, it's vital to transform them into forms that are easier to interpret, like interval notation or a number line graph. These methods allow us to visualize and understand the set of possible solutions effectively. Each form—algebraic, interval, or graphical—offers unique insights into what the inequality truly represents, helping students grasp the concept better.