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When it comes to inequality graphing, the primary goal is to visually represent the set of all possible solutions to an inequality on a number line. It's a powerful tool for understanding the range of values that satisfy the inequality. Let's take our example of the inequality \(x \leq -10\). The process begins with drawing a number line, which is an infinite straight line with numbers placed at equal intervals. To graph \(x \leq -10\), a solid circle is placed at the -10 mark on the line. This circle represents the number -10 itself, indicating that it is part of the solution set.

The next step is to shade the entire region to the left of -10. Shading this area signifies that every point in that region is a solution, which corresponds to all values of \(x\) that are less than or equal to -10. In inequalities where the variable does not include the number at the boundary (indicated by a < or > symbol), you would use an open circle to show that the number is not part of the solution.

Isolating the variable is a fundamental step in solving inequalities (and equations). This process involves moving all terms containing the variable to one side of the inequality and the constants to the other side. This can be done using basic algebraic operations like addition, subtraction, multiplication, and division, ensuring the operations are performed on both sides of the inequality to maintain balance.

In our exercise, we start with \(x + 7 \leq -3\) and aim to isolate \(x\). We subtract 7 from both sides which gives us the simplified form \(x \leq -10\). It's crucial to remember that if we ever multiply or divide by a negative number while isolating the variable, the direction of the inequality must be reversed to preserve the true relationship between the values.

The number line representation is an excellent way to express the solutions of inequalities. It converts abstract inequality statements into a clear visual format. In our example, we graph \(x \leq -10\) by first drawing a horizontal line, which represents our number line. This line needs to include the value mentioned in the inequality as well as sufficient range to clearly show which part of the line is shaded.

We utilize a solid dot to denote that -10 is included in the solution set. We then shade to the left of -10 to indicate all the values that \(x\) can take. This graphic aid not only assists students in deciding which values are part of the solution set but also helps to understand how different inequalities would be represented, such as distinguishing between 'less than' and 'less than or equal to' through open and closed dots respectively.