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Inequality graphing is a visual method to represent the range of solutions for an inequality on a number line, which is particularly useful for understanding the values that a variable can take.

For instance, when we want to solve the inequality \[\begin{equation}-8 < x-8 < 8\text{,}\end{equation}\] our first goal is to isolate the variable (a step we'll discuss in detail in the next section). Once we have the simplified inequality \[\begin{equation}0 < x < 16\text{,}\end{equation}\] we can then move on to the graphing process. This involves placing two boundary points on the number line, in this case, the numbers 0 and 16. Since 'x' is strictly greater than 0 and less than 16, open circles are placed on these numbers to indicate that they are not included in the solution set. The numbers between these two points are the solutions to the inequality, so we shade this region to illustrate which values 'x' can take.

This visual representation not only aids comprehension but also ensures accuracy when determining the set of possible solutions.

Isolating the variable is a fundamental step in solving inequalities and equations. It involves manipulating the inequality so that the variable of interest is on one side of the inequality sign and everything else is on the other side.

Using the exercise \[\begin{equation}-8 < x-8 < 8\text{,}\end{equation}\] the goal is to free 'x' from any additional terms. This is achieved through the 'do to one side, do to all sides' principle, which maintains the inequality's balance. To isolate 'x', we add 8 to all parts of the compound inequality, thus eliminating the -8 from the middle portion. The new form becomes \[\begin{equation}0 < x < 16\text{,}\end{equation}\] where 'x' stands alone within the inequality. The simplicity of this form allows us to easily graph the solution set, as well as comprehend the range of values 'x' can assume.

The number line representation is a crucial tool that transforms an abstract inequality into a clear visual guide.

Once an inequality is simplified, such as \[\begin{equation}0 < x < 16\text{,}\end{equation}\] we use a number line to illustrate the set of solutions. The number line is a straight line with increments representing numbers. To graph the inequality, we identify the critical points—in this case, 0 and 16. We place open circles at these points to signal that they are not part of the solution because the inequality does not include the endpoints (indicated by using '<' instead of '\leq'). Any value between 0 and 16 satisfies the inequality, so we shade the region between the open circles. This visual aid is an effective way for students to understand at a glance which values are solutions to the inequality.