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Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They are denoted by symbols such as ">", "<", "≥", and "≤". In the inequality \(3 \leq x \leq 7\), two inequality signs are used to express that \(x\) is a value that is within a specific range. This particular inequality states that \(x\) must be

• greater than or equal to 3
• less than or equal to 7

Understanding this expression requires recognizing that \(x\) can take any value from 3 up to 7, including the numbers 3 and 7 themselves. The inequality is a way to describe this set of potential values in a concise form.

A closed interval in mathematics is used to specify a range of numbers where both endpoints are included. In interval notation, a closed interval is denoted by square brackets, like so: \([a, b]\). This signifies that all numbers between \(a\) and \(b\), as well as the endpoints themselves, are included in the set. For the inequality \(3 \leq x \leq 7\), the closed interval is represented as \([3, 7]\).
This means that every number between 3 and 7, including 3 and 7, is part of the interval. Closed intervals are crucial in mathematical solutions because they clearly outline the maximum and minimum bounds for which a solution is valid. They are used to communicate explicitly the restriction on which values of \(x\) are covered.

Graphing inequalities involves visualizing an inequality on a number line to provide a clear, graphical representation of the solution set. The inequality \(3 \leq x \leq 7\) can be graphically represented on a number line in a few straightforward steps:

• Draw a horizontal line to represent the number line
• Mark the points 3 and 7 on this line
• Use solid dots at both 3 and 7 to illustrate that these endpoints are included (since it is \(\leq\) and \(\geq\))
• Shade the line segment that connects these two points, indicating all values between 3 and 7 are included in the solution

Graphing such inequalities is helpful because it provides a visual method to interpret and understand the range of possible solutions.

Number lines are simple yet effective tools for representing numbers, intervals, and inequalities. By using a number line, you can easily depict intervals in a format that is more comprehensible compared to algebraic expressions alone. For the inequality \(3 \leq x \leq 7\), the number line representation works as follows:

• Draw out a straight horizontal line, labeling it with numbers
• Identify and clearly label the endpoints, 3 and 7
• Place solid circles (or dots) at the points marking 3 and 7 to specify inclusion
• Draw a thick line or use shading between these marks to indicate the range of valid \(x\) values

A number line representation offers a straightforward and visual way for students to understand the concept of intervals and inequalities, showing exactly which numbers satisfy the inequality or belong to the interval.