91Ó°ÊÓ

To solve inequalities, we are looking to find the range of values for a variable that makes the inequality true. Differing from equations, inequalities do not show exact solutions but rather a set of potential solutions that satisfy the condition.
One key method is to carefully handle absolute value inequalities. When solving an absolute value inequality like \(|x-1| \leq 2\), it's crucial to break it into two separate inequalities to capture all possible values that\(|x-1|\) spans.
In general, for \(|A| \leq B\), we consider:

• \(A \leq B\)
• \(A \geq -B\)

By solving these inequalities separately, you determine the intervals where the original inequality holds true.
The solution process involves: evaluating each part separately, finding the collective solution, and then combining results to establish the overall range or solution set suitable for the original inequality.

Compound inequalities are composed of two separate inequalities linked by the words "and" or "or". In absolute value inequalities, such as those explored in this exercise, **compound inequalities with an 'and' relationship** indicate that a solution must satisfy both conditions at the same time.
For instance, in the inequality \(|x-1|\leq 2\):

• The 'and' operation is apparent where \(x - 1 \leq 2\)
• And \(x - 1 \geq -2\) are both satisfied

These are evaluated separately but must both "agree" on the values for \(x\).
When dealing with compound inequalities, it’s important to determine whether you are intersecting two sets or finding their union. This process ensures that solutions like \(-1 \leq x \leq 3\) from this exercise account for the requirements of both constituent inequalities.

The final step in solving absolute value inequalities is to interpret the result, which is an inequality solution. The solution represents an interval or set of values that satisfies the original absolute value inequality.
For \(|x-1| \leq 2\), we've found the solution \(-1 \leq x \leq 3\), which implies that any value of \(x\) between -1 and 3, inclusive, will make the original inequality true.
It's imperative to use interval notation or graphing to clearly present solutions:

• The solution \(-1 \leq x \leq 3\) can be written in interval notation as \([-1, 3]\).
• This notation represents all numbers \(x\) such that \(x\) is greater than or equal to -1 and less than or equal to 3.

By translating the solution into different forms, you ensure clarity, allowing others to easily grasp the range of potential values \(x\) can take to satisfy the inequality. Similarly, graphing the solution on a number line often provides a visual understanding of the range of values presented by the solution.