91Ó°ÊÓ

Compound inequalities involve more than one inequality statement joined by the word 'and' or 'or'. In the given problem, the compound inequality \(-11 < 2x - 1 \leq -5\) can be separated into two individual inequalities:

• \(-11 < 2x - 1\)
• \(2x - 1 \leq -5\)

These two inequalities are connected by 'and', which means the solution must satisfy both conditions simultaneously. The approach requires solving each inequality on its own and then finding the overlap of the solutions. This combination forms the final solution of the compound inequality.

Graphing inequalities on a number line helps visually display the solution set. Once you have solved each inequality separately, you can represent the solution on a number line. For compound inequalities, you need to consider the overlap between the solution sets, because they are connected by 'and'. In this example, once you solve:

• \(-5 < x\)
• \(x \leq -2\)

You determine that the solution for the compound inequality is where \(x\) is greater than \(-5\) but less than or equal to \(-2\). To represent this on a number line:- Use an open circle at \(-5\) to show that \(-5\) is not included in the set.- Use a closed circle at \(-2\) to indicate that \(-2\) is included.- Shade the region in between which signifies the values that meet both conditions.

Graphing inequalities provides a visual way to understand which values satisfy the inequalities. This step is crucial for compound inequalities as it clearly shows the intersection of solutions from individual inequalities. When graphing, you must pay attention to:

• Open circles, which indicate that a number is not included in the solution set.
• Closed circles, meaning the number is included.
• The shaded regions, which represent all numbers that satisfy the inequalities in the solution set.

In this exercise, you will plot an open circle at \(-5\), a closed circle at \(-2\), and shade the area between them, showing all \(x\) values that are greater than \(-5\) and less than or equal to \(-2\). Graphing helps confirm the logical solution and provides another layer of understanding.