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Understanding how to solve inequalities is a fundamental skill in math that allows students to find the range of possible values for variables within an equation. Inequalities differ from equations as they do not imply exact equality, but rather express a relationship where one quantity is either less than, greater than, less than or equal to, or greater than or equal to another. When solving an inequality, such as \(24x < 100\), the goal is to isolate the variable on one side.

To solve this, divide both sides by the constant that multiplies the variable, in this case, 24. By doing so, you alter the inequality to \(x < \frac{25}{6}\), which provides a clearer view of the range of values that \(x\) can take. It's crucial to remember that if you ever multiply or divide by a negative number while solving an inequality, you must flip the inequality symbol to maintain the true relationship between the two sides. After solving, you can plot the values on a number line or enumerate specific solutions if \(x\) is restricted to integers or natural numbers.

Natural numbers are the set of positive integers that begin with 1 and increase incrementally by 1 forever: 1, 2, 3, 4, 5, and so on. They are the numbers we naturally count with and therefore do not include zero or negative numbers. In the context of inequalities, when an exercise requires the solution to be a natural number, you look for whole numbers greater than zero that satisfy the inequality.

For example, after solving \(24x < 100\), we found that \(x < \frac{25}{6}\) or \(x < 4.16666\ldots\). Considering only natural numbers, the solution is limited to 1, 2, 3, and 4, which are all the natural numbers less than \(\frac{25}{6}\). Highlighting natural numbers as part of the exercise ensures the understanding that decimals and fractions can't be solutions in this context.

Integers are a broader category than natural numbers, encompassing all whole numbers and their negatives, including zero: ..., -3, -2, -1, 0, 1, 2, 3, .... They are the building blocks of our number system for counting, ordering, and arithmetic, extending infinitely in both the positive and negative directions. When we direct our attention to solving inequalities for integers, we must include the negative whole numbers and zero in our solutions.

In an inequality like \(24x < 100\), where \(x\) is an integer, the solution includes all integers that are less than 5 due to \(4.16666\ldots\) being the fractional boundary. Therefore, 4, 3, 2, 1, 0, and all negative whole numbers are part of the solution. When exercises specify that \(x\) is an integer, it is signaling to consider the full range of whole numbers, not just the positive side illustrated by natural numbers.