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Interval notation is a convenient way to represent the range of solutions to an inequality. It uses parentheses \(\text{( )}\) and brackets \[\text{[ ]}\] to indicate whether endpoints are included or excluded.
For example, when we have the solution \(x \leq 6\), it means that \(x\) can be 6 or any number less than 6, going all the way to negative infinity. We represent this as:

• \((-\infty, 6]\) - Here, \(-\infty\) is always expressed with an open parenthesis because infinity is not a specific number that can be "included."
• The number 6 is written within a closed bracket because it is part of the solution set.

Remember, in interval notation, **open parentheses** are used when the endpoint is not included (\(<\) or \(>\)), and **closed brackets** indicate the endpoint is included (\(\leq\) or \(\geq\)).

The solution set of an inequality is all the values that satisfy the inequality condition. It's like the answer to a puzzle: all the x-values that make the original inequality true.
In the problem at hand, we simplified the inequality \[\frac{x-7}{2}-\frac{x-1}{5} \leq-\frac{x}{4}\] into a simpler inequality \[x \leq 6.\]This tells us that any number \(x\), as long as it is six or less, will make the inequality true when substituted back into the original expression.

• A solution set can be many numbers, a single number, or even no numbers (when no real values satisfy the inequality).
• Here, all numbers from negative infinity to 6, inclusive, make up the solution set.

The idea of a solution set helps verify if a solution is correct and whether it actually fits within the parameters of the problem.

Graphing solutions for inequalities on a number line visually represents the solution set.
Consider it a mini-map guiding the user through the realm of all possible solutions. Here's how you can graph the inequality \(x \leq 6\):

• First, draw a horizontal line to denote the number line.
• Mark the number 6 on this line. Since our inequality \(x \leq 6\) includes the number 6, place a closed circle (or dot) directly on this point.
• From this closed circle, shade or draw an arrow extending to the left towards negative infinity to show that all numbers less than 6 are also solutions.

This visual representation aids in checking and confirming the solution quickly, especially when comparing different solutions or inequalities.

Finding a common denominator is a crucial first step in solving inequalities involving fractions. It allows you to eliminate these fractions and simplify the process.
In the given example:

• We have fractions with denominators of 2, 5, and 4.
• The least common denominator (LCD) for these numbers is 20.

By multiplying each term of the inequality by 20, we successfully clear out the fractions, transforming the inequality into a simpler form without needing to deal with fractional parts:\[20 \cdot \left(\frac{x-7}{2}\right) - 20 \cdot \left(\frac{x-1}{5}\right) \leq 20 \cdot \left(-\frac{x}{4}\right)\]This step is essential because simplifying the inequality makes subsequent steps (e.g., distributing terms and combining like terms) much easier to execute.
Having a clear path to a simpler equation helps ensure we solve the inequality efficiently and accurately.