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Inequalities allow us to express a range of possible values for a variable, not just a specific one. In mathematics, an inequality shows that two expressions are not equal and involves comparison operators such as ">", "<", "≥", and "≤". Each symbol carries a specific meaning:

• "<" means less than
• ">" means greater than
• "≤" means less than or equal to
• "≥" means greater than or equal to

Compound inequalities consist of two or more inequalities joined together by either "and" or "or". An "and" compound inequality means that both conditions must be satisfied simultaneously, creating an intersection of solutions. On the other hand, an "or" compound inequality only needs one condition to be true, resulting in a union of solutions. To solve an inequality, the process is quite similar to solving an equation, with a special rule for multiplication or division by negative numbers, which flips the inequality sign. This fundamental concept is crucial since it allows us to determine ranges of solutions rather than exact answers.

After solving an inequality or a compound inequality, we often express the solution using interval notation. This notation is a concise way of showing which parts of the number line are included in the solution. Interval notation uses brackets and parentheses:

• "[" or "]" indicates that an endpoint is included (closed interval)
• "(" or ")" shows that an endpoint is not included (open interval)

For example, if the solution is all numbers greater than or equal to 5, we write it as \([5, \, \infty)\). The square bracket signifies that 5 is part of the solution, and the parenthesis around infinity reflects that infinity is an idea of boundlessness - we can approach it but not include it. Interval notation is a powerful tool for summarizing the results of inequalities in a clean and efficient form, and it's especially helpful when graphing solutions on a number line.

Graphing inequalities on a number line gives a visual representation of the solutions, offering a clear and intuitive way to understand the range of possible values. On a number line, the solution to an inequality is typically shown with either a filled or an open dot and a shaded region:

• A filled dot indicates the number is included in the solution (≥ or ≤)
• An open dot shows the number is not included (> or <)

Once the endpoints are decided, the region extending in the direction of the inequality is shaded. For instance, the inequality \(x \geq 5\) is represented by a filled dot at 5 and shading to the right, showing all numbers from 5 to infinity are included. Graphing on a number line is particularly handy for visualizing the solution's scope and is often used alongside interval notation to provide a comprehensive view of inequality solutions.