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Inequalities are mathematical expressions used to compare two values or expressions. An inequality shows that one value is either less than, greater than, or equal to another value, excluding values where they are exactly equal, unless stated otherwise. In the given inequality, \(-1 \leq s \leq 2\), it uses both "less than or equal to" (\(\leq\)) symbols. This means:

• \(-1 \leq s\) implies that \(s\) can be any number that is equal to \(-1\) or greater than \(-1\).
• \(s \leq 2\) indicates that \(s\) can also be equal to or less than \(2\).

Together, the inequality states that \(s\) includes all numbers from \(-1\) to \(2\), inclusive of the endpoints. Inequalities help determine a range of possible values and are foundational in various areas of math like calculus, algebra, and optimization problems.

When graphing such inequalities, it is important to recognize whether endpoints are included or excluded. Here, both \(-1\) and \(2\) are included, reflected by using "closed intervals," which are signified using brackets \([-1, 2]\).

A real number line is a straight line on which real numbers can be represented as points. It’s a convenient way to visualize equations and inequalities, allowing for easy comparison between numbers. For the inequality \(-1 \leq s \leq 2\), the real number line serves as the perfect tool to identify the solution set.

• First, draw a horizontal line and mark major points that are significant to the problem, such as \(-1\) and \(2\).
• Both points \(-1\) and \(2\) are marked clearly, often with a filled or darkened dot to show they are inclusive in the solution.
• The line interval between two points is shaded or highlighted to indicate all values \(s\) could take.

Using a real number line helps in understanding and verifying the range represented by inequalities; thus, improving comprehension and ensuring accuracy when solving mathematical problems.

The solution set of an inequality consists of all the values that make the inequality true. For an inequality like \(-1 \leq s \leq 2\), the solution set includes every potential number \(s\) that satisfies the condition of being equal to or greater than \(-1\), and equal to or less than \(2\).

• This means integers like \(-1, 0, 1,\) and \(2\) are definitely included.
• In fact, any real number in this range, whether whole, fractional, or decimal, fits within the solution set.
• The representation of inclusive inequalities on a number line, where the end points are shaded, signifies that endpoints are part of the solution set.

Graphing allows us to visually affirm the solution set, supporting our understanding of which values are possible. In practical terms, it informs us of all the outcomes that satisfy a given condition, providing a clear and concise method to express mathematical relationships.