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Interval notation is a concise way of describing a set of numbers that fall between a lower bound and an upper bound on a number line. When using interval notation, parentheses "(" or ")" indicate that a particular endpoint is not included in the set (strict inequality), while square brackets "[" or "]" indicate inclusion (non-strict inequality).

For example, in the inequality \(-3 < x \leq 1\), you have:

• A parenthesis "(" at \(-3\) to show that it is not part of the solution set — meaning \(-3\) must not equal \(x\) (strict inequality).
• A bracket "]" at \(1\) to indicate that \(1\) is included in the solution set — meaning \(-3 < x \leq 1\) does allow \(x\) to equal \(1\) (non-strict inequality).

Therefore, this time interval is expressed as \((-3, 1]\), capturing all values up to \(1\) and beyond \(-3\), but not including \(-3\) itself.

Graphing inequalities on a number line provides a visual representation of the solution set for inequalities, helping to illustrate which numbers satisfy the inequality condition.

When plotting inequalities, different types of circles are used:

• Open circles represent values that are not included in the inequality, as in the point \(-3\) in our inequality \(-3 < x \leq 1\). This is because the inequality at \(-3\) is strict ("<"), so \(-3\) itself isn't part of the solutions.
• Closed circles show that the value is included in the inequality, which is the case for \(1\) in \(-3 < x \leq 1\). The closed circle at \(1\) indicates it is part of the solution, agreeing with the non-strict inequality ("\leq").

Additionally, shading or a solid line is drawn between these distinct points to denote all values between and satisfying the inequality, allowing you to see all possible solutions at a glance.

Understanding the difference between strict and non-strict inequalities is key to solving and graphing inequalities correctly. A strict inequality uses symbols like "<" or ">", indicating that the number is "less than" or "greater than" another number but does not include that number.

For instance, in the inequality \(-3 < x\), \(-3\) is not a possible value for \(x\) because it's not included — this is marked with an open circle on the graph. On the other hand, non-strict inequalities utilize "\leq" or "\geq", where the value can be equal to the boundary number. This is represented in graphing by a closed circle.

Thus, in \(x \leq 1\), \(1\) can indeed be a value for \(x\). Recognizing which part of the inequality is strict or non-strict determines how you express and solve it accurately.