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A number line is a visual representation of numbers placed in order on a straight line. Think of it like a ruler where you can measure the value of numbers. To sketch inequalities, start by drawing a horizontal line and mark several points to represent numbers. It's important to include the values relevant to the inequality being graphed, as well as numbers before and after, to give context to where the inequality lies in the overall set of real numbers.

For example, to graph the inequality \(x \geq 6\), you would first draw a straight line and then evenly space out marks for numbers, ensuring 6 is clearly plotted. You would label enough numbers both to the left and right of 6 to provide a comprehensive view of where 6 stands in relation to other numbers. This visual aide immensely helps students comprehend the range of values that \(x\) may take.

Inequality notation provides a succinct way to write the relationship between numbers. It uses symbols to convey if one number is greater than, less than, or equal to another number. For example, \(\gt\), \(\lt\), \(\geq\), and \(\leq\) stand for 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to', respectively.

The expression \(x \geq 6\) is read as 'x is greater than or equal to 6'. Here, the 'equal to' part is crucial because it specifies that the value of \(x\) can also be exactly 6, not just numbers larger than 6. Inequalities can describe many real-world situations, such as establishing boundaries. A deep understanding of inequality notation is essential to correctly interpret and represent mathematical statements involving inequalities.

When graphing inequalities on a number line, the type of circle used at the plotted number conveys different meanings. A closed circle, or filled in dot, indicates that the number it's placed upon is included within the set of solutions to the inequality. For the inequality \(x \geq 6\), you would put a closed circle on the number 6 to show that 6 is a solution to the inequality; it's part of the set.

A closed circle is essential for inequalities that use \(\geq\) or \(\leq\). On the contrary, if the inequality did not include the number – if it used \(\gt\) or \(\lt\) instead – you would use an open circle, which signifies that the number itself is not included in the solution set but that the solution is any number greater or less than the one marked. Understanding the difference between a closed and an open circle eliminates confusion when interpreting graphed inequalities.