91Ó°ÊÓ

An absolute value inequality involves an expression wrapped in absolute value bars, such as \(|x^2 - 5| \geq 4\). Absolute value measures how far a number is from zero, regardless of direction. This means something like \(|a| \geq b\) translates to considering two cases: either the expression inside is greater than or equal to \(+b\) or less than or equal to \(-b\).
\[ \ |x^2 - 5| \geq 4 \text{ implies two scenarios: } \ x^2 - 5 \geq 4 \text{ or } x^2 - 5 \leq -4. \]
This double condition stems from the nature of absolute values encompassing both positive and negative possibilities. In our solved example, this resulted in two distinct expressions: \((x^2 - 5 \geq 4)\) simplifies to \((x^2 \geq 9)\), while \((x^2 - 5 \leq -4)\) simplifies to \((x^2 \leq 1)\). Understanding this dual-condition is crucial to mastering absolute value inequalities.

Quadratic inequalities involve expressions with a degree of two, meaning the variable is squared (e.g., \(x^2 \geq 9\) or \(x^2 \leq 1\)).

Here's how we manage such inequalities:

• Identify opportunities to take the square root on both sides.


• This operation will yield two potential outcomes indicating ranges of solutions.


• For example, solving \(x^2 \geq 9\) gives \(x \geq 3\) and \(x \leq -3\).
• Conversely, for \(x^2 \leq 1\), we have \(-1 \leq x \leq 1\).

The difference between these two kinds of inequalities lies in whether they describe an interval (like \(-1 \leq x \leq 1)\) or two separate branches (\