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A compound inequality involves two separate inequalities joined by either "and" or "or". In our example, the compound inequality is \[-5 < 2(x-1) - 3(x+2) < 5\] This combines two inequalities in one statement. The solution must satisfy both individual inequalities simultaneously. In other words, a number is a solution only if it works for both inequalities at the same time. This is different from a simple inequality, which contains only one comparison to deal with.

• "And" implies overlap - the solutions are in the region where both inequalities are true.
• "Or" implies union - solutions are valid if they satisfy at least one of the inequalities.

In this problem, we are addressing a type of "and" compound inequality, as the variable must satisfy both parts to remain in the range of valid values.

Interval notation is a concise way to express the solution to an inequality. In this case, after solving the inequality, we got: \[-13 < x < -3\] This is expressed as \((-13, -3)\) in interval notation. The round brackets indicate that -13 and -3 are not included in the solution (known as open intervals).

• Round brackets \(( )\) signify numbers that are not part of the solution.
• Square brackets \([ ]\) mean a number is included in the solution.

Interval notation is helpful because it's clear, simple, and avoids the need for inequality symbols when writing the range of solutions.

Graphing solutions on a number line is a visual way to represent the range of numbers that satisfy an inequality. For \((-13, -3)\), we depict this interval with open circles at -13 and -3, signaling these points are not included in the solution. Draw a line connecting these points to show that all numbers between -13 and -3 are part of the solution set. This graphical representation helps visualize the extent and nature of solutions:

• Open circles indicate that endpoints are not included in the solution.
• A connected line shows all numbers within the interval are valid solutions.

Using a number line can simplify understanding complex inequalities, especially when combined with interval notation.

Isolating the variable is a critical step in solving inequalities. The goal is to have the variable, in this case, \(x\), alone in the center of the inequality. We started with the expression: \[-5 < -x - 8 < 5\]. We can isolate \(x\) by executing actions like adding, subtracting, multiplying, or dividing across every part of the inequality. Crucially, remember to reverse the inequality symbol when multiplying or dividing by a negative number. Here’s how we did it:

• Add 8 to each segment, moving towards isolating \(-x\).
• Multiply through by -1, flipping inequality signs, to solve for positive \(x\).

This process transforms the original inequality into a simpler form: \[-13 < x < -3\]. This technique applies broadly across algebraic problems ensuring the variable is cleanly isolated in its most readable position.