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Inequalities are mathematical expressions that compare two quantities. They tell us whether one value is larger than, smaller than, or simply not equal to another value. In the case of an inequality like \(0 < x < 5\), it conveys that the variable \(x\) is any number greater than 0 and less than 5.

• "\(<\)" and "\(>\)" are symbols that represent "less than" and "greater than", respectively.
• "\(\leq\)" and "\(\geq\)" are used for "less than or equal to" and "greater than or equal to".

When you see an inequality, remember that it resembles a number line that hasn’t been drawn yet, with points marked to show which numbers satisfy the condition described. Understanding inequalities helps in identifying the range within which a variable can change.

Graphing intervals is a way of visually representing inequalities on a number line. It helps in clearly understanding the range of values that a variable can take. Here's a step-by-step guidance for graphing an interval such as \( (0, 5) \):

• First, draw a horizontal line. This represents the number line.
• Mark important points (like 0 and 5) along the line. These are your boundary points of the interval.
• Use open circles (hollow dots) to mark 0 and 5, indicating that they are not included in the interval.
• Shade the entire line segment between these two open circles. This shaded part represents all the numbers \(x\) can be.

This graphical method makes it easier to visualize which parts of the number line the inequality refers to, ensuring there's no ambiguity about which numbers are included or excluded.

The number line is a simple yet powerful tool that helps in visualizing numbers and expressions involving numbers, such as inequalities and intervals. It extends infinitely in both directions and is equally spaced, marking every integer.

• Positive numbers are plotted to the right of zero.
• Negative numbers are plotted to the left of zero.

When working with intervals, it's important to understand that the number line provides a continuous set of possible values. Marking points and shading areas on the number line gives a clear idea of which numerical values a variable like \(x\) can take. The practice of moving left or right from zero helps in comprehending how quantities increase or decrease in value, creating a visual understanding that is crucial when dealing with more complex mathematical concepts.