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Compound inequalities involve two or more inequalities that are combined into one statement. In the context of absolute value inequalities, such as \(|x-2|<4\), the inequality can be split into a pair of separate linear inequalities.

When dealing with an absolute value inequality like \(|x-a|

\(x-a < b\)

and \(x-a > -b\).

This set of compound inequalities can often be written less formally in a compact form like

• \(-b < x-a < b\).

This expression captures the range of potential \(x\) values that satisfy the original absolute value inequality. Solving these compound inequalities will pinpoint the exact range of solutions.

Graphing inequalities on a number line helps to visually represent the solution set of the inequality. For the compound inequality

• \(-2 < x < 6\)

this illustrates that \(x\) can be any number that lies between \(-2\) and \(6\), but not including these end values.

To show this on a number line:

• Draw an open circle around \(-2\) to signify that \(-2\) is not a solution.
• Do the same for \(6\), also with an open circle.
• Shade the region in between these two numbers to denote all the potential values for \(x\) in this range.

These visual cues play a crucial role in understanding exactly which numbers satisfy the inequality. Whenever you have an inequality, putting it on a number line helps clarify and confirm the solution set visually.

Solving an inequality requires careful step-by-step manipulation to ensure that the solution set is determined accurately. Let's consider the problem

• \(|x-2|<4\).

First, rewrite this absolute value inequality as a compound inequality:

• \(-4 < x-2 < 4\).

This restates the problem in a form that is solvable using linear inequality steps.

Next, solve each part of the compound inequality separately. For the left part,

• \(-4 < x-2\), which becomes \(-4 + 2 < x\).
• This simplifies to \(-2 < x\).

For the right part,

• \(x-2 < 4\), adding 2 gives \(x < 6\).

Together, these inequalities show that the solution is all numbers \(x\) such that

• \(-2 < x < 6\).

Each step in solving the inequality is essential for isolating the variable and finding the complete set of solutions. Careful execution of these steps ensures accurate and reliable results for the inequality.