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Graphing inequalities is a visual representation of all possible solutions to an inequality. It involves plotting on a number line or on a coordinate plane, depending on whether you're dealing with one variable or two.

In the case of the inequality given in the exercise, \(9 \geq -15 + x\), the graphing is one-dimensional since it involves only one variable, \(x\). After solving the inequality, we have \(x \leq 24\), meaning that the value of \(x\) can be 24 or any number less than 24.

To represent this graphically, a number line is used, which is a horizontal line with numbers placed at equal intervals. For the inequality \(x \leq 24\), we diagram a closed circle at \(24\) to show that \(24\) is included in the set of solutions (since the inequality allows for \(x\) to be equal to \(24\)). An arrow is then drawn extending to the left, to indicate all numbers less than \(24\) are also included in the solution set. This graphical approach provides an immediate visual understanding of the range of possible values for \(x\).

Isolating variables is a fundamental step in solving equations and inequalities. It involves rearranging the equation or inequality so that the variable you are solving for is on one side by itself.

In our example, \(9 \geq -15 + x\), we want to isolate \(x\). To accomplish this, we perform the same operation on both sides of the inequality, which maintains the equality or inequality relationship. We add \(15\) to both sides, resulting in \(24 \geq x\), which can also be written as \(x \leq 24\).

This step is crucial as it simplifies the problem, making it easier to understand and solve. The key is always doing the same manipulative action to both sides to keep the inequality balanced. It's just like balancing scales. If you add weight to one side, you must add the same weight to the other side to keep it balanced.

Number line representation is a great tool for visualizing and solving inequalities. A number line is a straight line with numbers placed at intervals, which helps to see the relative position of numbers.

When working with an inequality, such as \(x \leq 24\) from the given example, you can easily illustrate the set of numbers that satisfy the inequality on a number line. Here’s how you do it: draw a straight horizontal line, mark the point at \(24\), and since the inequality includes \(24\), darken the circle at \(24\) to indicate that it is part of the solution (known as a closed dot). Next, from the point \(24\), draw an arrow that extends leftward to show that all numbers to the left of \(24\) (those that are less than \(24\)) are also solutions to the inequality.

This representation provides a straightforward way to see the range of viable solutions and is particularly helpful for students to understand the concept of inequalities and their solutions.