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A compound inequality involves two separate inequalities that are joined by either "and" or "or." In the case of absolute value inequalities like \(|x+1| \leq 1\), this can be converted into a compound inequality by recognizing what the absolute value represents.

A key point to understand is that the absolute value \( |x+1| \) signifies the distance from zero and is always positive. Given \(|x+1| \leq 1\), it means that whatever \(x+1\) is, its distance on the number line from zero is at most 1.

This transforms into a compound inequality: \(-1 \leq x+1 \leq 1\).

• It translates to saying that \(x+1\) must be greater than or equal to -1 and less than or equal to 1.
• So, we solve each part: \(-1 \leq x+1\) and \(x+1 \leq 1\).

Next, let's solve these two inequalities to find the possible values of \(x\).

Graphing inequality solutions on a number line is a visual way to represent solutions. Once we have figured out our solution from the compound inequality, such as \-2 \leq x \leq 0\, the next step is plotting them on a number line.

First, identify the critical values from the inequality. Here, they are -2 and 0. These are the endpoints of your interval on the number line.

• A closed dot is used to indicate that the point is included in the solution set (e.g., \(-2,0\) are included because our inequality has "less than or equal to").
• Draw a solid line connecting the numbers -2 and 0 to indicate that all numbers in between are part of the solution.

Visualizing on a number line helps in understanding the range and limits of solutions, showing clearly where \(x\) can possibly fall.

Solving absolute value inequalities involves systematic steps to ensure a correct solution. These steps help in breaking down the inequality into simpler parts:

1. **Understand the Absolute Value**:
Recognize how the absolute value affects the terms. Here, \(|x+1|\leq1\) is previewed as considering the distance of \(x+1\) from zero.

2. **Rewrite as a Compound Inequality**:
Convert \(|x+1| \leq 1\) into \-1 \leq x+1 \leq 1\. This process uses the definition of absolute value to eliminate it.

3. **Solve Each Part Separately**:

• For the left inequality: Solve \-1 \leq x+1\ by subtracting 1 from both sides, leading to \-2 \leq x\.
• For the right inequality: Solve \x+1 \leq 1\ by subtracting 1 from both sides, resulting in \x \leq 0\.

4. **Combine the Solutions**:
With both parts solved, integrate them into one combined solution, like \-2 \leq x \leq 0\.

Following these steps ensures clarity, reduces errors, and helps in reliably finding solutions.