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Interval notation is a method of writing down a range of numbers that are solutions to an inequality. It gives a clear and concise picture of the solution set. When you solve an inequality like the one in our exercise, you often end up with a range of acceptable values for the variable involved.

To express this range, interval notation is used:

• A round bracket '(' or ')' indicates that the endpoint is not included in the interval, which corresponds to a less than "<" or greater than ">".
• A square bracket '[' or ']' indicates that the endpoint is included, which corresponds to a less than or equal to "≤" or greater than or equal to "≥".

In this exercise, we found that the solution to the inequality is \[ x \leq \frac{15}{4} \]This means that \( x \) can be any number less than or equal to \( \frac{15}{4} \).

The interval notation for this solution is expressed as:\[ (-\infty, \frac{15}{4}] \]Here, \(-\infty\) denotes that there is no lower boundary, and \(\frac{15}{4}\) is the point at which x is capped, inclusive of \(\frac{15}{4}\). By understanding interval notation, you can better interpret and communicate solutions to inequalities effectively.

Graphing inequalities involves visually representing the solution set on a number line, which can help in understanding the range of values that satisfy the inequality. To illustrate the solution of an inequality, like \[ x \leq \frac{15}{4} \]you would use a number line.

Here are the steps for graphing this particular inequality:

• First, find the numerical point on the number line that corresponds to \(\frac{15}{4}\), which is equivalent to 3.75.
• Place a solid dot at this point to indicate that it is included in the solution set (due to the "≤" in our inequality).
• Shade the number line to the left of \(\frac{15}{4}\) to demonstrate that all numbers less than \(\frac{15}{4}\) are included in the solution.

By doing so, you are providing a clear visual representation of where the solution lies, making it easier to interpret the possible values \( x \) can take based on the given inequality. This technique also allows for quick verification if a particular value satisfies the inequality.

Solving inequalities algebraically often involves similar methods as solving equations, but with some key differences. In our exercise, we begin with the inequality:\[\frac{5}{9}(x+3)-\frac{4}{3}(x-3) \ge x-1\]The steps to find the solution include:

• Distributing each fraction over the terms inside the parentheses, which is a critical step to simplify the expression.
• Combining like terms and ensuring all terms involving \(x\) are on one side of the inequality.
• To isolate \(x\), you may need to perform operations like adding or subtracting the same value from both sides, or multiplying/dividing by a constant.

In our exercise, after simplifying and isolating \(x\), we arrive at \[ x \leq \frac{15}{4} \].Remember, when multiplying or dividing by a negative number, the inequality sign must be flipped. Solving algebraically not only gives us a precise answer but equips us with the understanding necessary to translate these calculations into interval notation and appropriate graphs.