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When we talk about solving inequalities, we are referring to the process of finding all the possible values of a variable that make the inequality statement true. In an inequality, instead of an 'equals' sign, you'll see symbols like '<' (less than), '>' (greater than), '(leq)' (less than or equal to), or '(geq)' (greater than or equal to). These symbols set the relationship between the expressions on either side.

Let's take the compound inequality from our exercise involving two separate inequalities: '(6x (geq) -24)' and '(9x < 54)'. To solve these, we isolate the variable 'x' in each inequality. For '(6x (geq) -24)', we divide both sides by 6 to get '(x (geq) -4)'. Similarly, for '(9x < 54)', dividing by 9 yields '(x < 6)'. It's crucial to keep the inequalities' directions correct while you manipulate the expressions, especially when multiplying or dividing by negative numbers.

After solving inequalities, graphing them is a visual way to represent the range of possible solutions. A number line is a straight line with numbers placed at intervals, which can show us where the solutions lie in relation to each other.

For the compound inequality '(-4 (leq) x < 6)', we use two different symbols on the number line. A solid circle at -4 indicates that -4 is part of the solution (since the symbol is '(leq)'), and an open circle at 6 indicates that 6 is not part of the solution (since the symbol is '<'). The shaded region in between these marks shows all the numbers that are solutions to the inequality, capturing every number greater than or equal to -4 and less than 6.

Using a number line representation is an excellent way to clearly demonstrate the solution to compound inequalities. It provides a simple visual summary of all acceptable values. Numbers that are included in the solution are typically marked with solid dots or lines, while exclusions are represented with open dots or dashed lines. This emphasizes the continuity of the solution set for an inequality.

For example, in our compound inequality, graphing it on a number line makes it instantly clear to anyone viewing it where the solutions lie. Not only does this method help confirm the accuracy of our algebraic solutions, but it can also be a quicker way to check solutions in multiple choice questions or while cross-referencing your work.