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Understanding how to graph absolute value inequalities is a crucial skill in mathematics. The absolute value of a number is its distance from zero on the number line, regardless of direction. So, it's always a non-negative number. When graphing, you'll typically be dealing with two scenarios – where the absolute value is less than a number (creating an 'inside' solution set) or greater than a number (an 'outside' solution set).

For the exercise at hand, after isolating the absolute value expression, we have to consider both less than and greater than scenarios. Graphing these on a coordinate plane generally involves V-shaped graphs opening upwards or downwards. However, with inequalities, we're focused on a one-dimensional number line. You'll mark the critical points – the values that make the inside of the absolute value zero – and shade either inside or outside these points depending on the inequality. In this case, the inequality solution, consisting of all numbers between 4 and 5, is indicated by an open circle at these points and shading in-between, reflecting that these endpoints are not included.

Solving inequalities differs from equations as the solution is often a range or set of numbers rather than a single value. When the inequality involves an absolute value, it usually results in two separate inequalities, without the absolute value, to consider. It's a pivotal step, as it addresses the 'two-sided' nature of absolute values – reflecting the distance on both sides of zero on the number line.

In our example, we first balanced the inequality by isolating the absolute value on one side, leading to a comparison with 1. From there, we created and solved two linear inequalities, resulting in the solution set where x is between 4 and 5. Remember, understanding the rules of inequality manipulation is vital. Reversing the inequality sign when multiplying or dividing by a negative number is one such important rule to keep in mind when finding your solution.

Representing solutions to inequalities on a real number line allows us to see the set of all possible solutions at a glance. A number line is a visual tool that shows numbers as points on a line, with the position based on their value. For the absolute value inequality we are examining, the solution is shown as a segment on the number line instead of discrete points.

In practice, an open circle is used to denote that a number is not included in the set (denoting 'less than' or 'greater than'), while a closed circle signifies that the number is included ('less than or equal to' or 'greater than or equal to'). A common mistake is to forget to flip the inequality sign after multiplying or dividing by a negative number during the solution process. In our exercise, the number line will display open circles at 4 and 5 with a shaded region between them, symbolizing all real numbers greater than 4 and less than 5—a handy representation of the range of solutions to the inequality.