91Ó°ÊÓ

Absolute value inequalities are a bit different from regular inequalities. The absolute value of a number is its distance from zero on the number line, regardless of direction. In math, it's represented as \(|A|\) where \(|A|\) is always positive or zero.
When we have an inequality like \( |x+2| > 1 \), it means that the value inside the absolute value brackets (\( x+2 \)) is more than 1 unit away from zero, in either direction.
This gives us two possibilities: either \( x+2 \) is greater than 1 (meaning it's more than 1 unit to the right of zero), or \( x+2 \) is less than -1 (meaning it's more than 1 unit to the left of zero).
So, we translate \( |x+2| > 1 \) into two separate inequalities: \( x + 2 > 1 \) and \( x + 2 < -1 \). Solving these inequalities separately will give us the solution set for the original problem.

After solving inequalities, it's important to represent the solution set clearly. Interval notation is a method used to denote a range or set of numbers along the number line. This form of notation is compact and easy to understand.
For this exercise, we found two conditions for \( x \: \).

• First inequality: \( x > -1 \)
• Second inequality: \( x < -3 \)

In interval notation, we represent these separately obtained solutions as intervals:
For \( x < -3 \), the interval is \( (-\backslash \backslash infty, -3) \).
For \( x > -1 \), the interval is \( (-1, \backslash \backslash infty) \).
Since these are two separate intervals (i.e., there are two non-overlapping ranges), we use the union operator (\( \backslash \backslash cup \)) to combine them. Therefore, the final answer in interval notation is \( (-\backslash \backslash infty, -3) \backslash \backslash cup (-1, \backslash \backslash infty) \).

Compound inequalities occur when two separate inequalities are joined by the word 'and' or 'or'. They describe a range of values for the variable that satisfy both inequalities.
There are two main types of compound inequalities:

• 'And' compound inequalities, which represent values that satisfy both conditions. For example, \(-3 < x < 1\) translates to \(-3 < x \) 'and' \( x < 1\).
• 'Or' compound inequalities, which represent values that satisfy at least one of the conditions. For example, \(x < -3 \) 'or' \( x > -1 \) in our exercise means any value less than -3 or greater than -1 is part of the solution set.

For the given exercise, \( 1 < |x+2| \) translates into \(x + 2 > 1 \) or \( x + 2 < -1 \), leading us to the compound inequality \( x > -1 \) 'or' \( x < -3 \). Using the word 'or' here is crucial as it suggests that satisfying either condition is sufficient for the solution.