91Ó°ÊÓ

When we're tasked with solving inequalities, like in this problem, we aim to find all values of a variable that make the inequality true. In our case, the inequality is given by \(x^4 - 1 \geq 0\). This means we're looking for all \(x\) values where the expression \(x^4 - 1\) results in zero or a positive number.
To tackle this, we start by factoring the expression, if possible. Once factored, the inequality can then be analyzed by determining where each factor is zero (these are the critical points) and testing intervals between these points to see where the original inequality holds true.
This approach allows us to break complex inequalities into manageable pieces, ensuring that we accurately determine the range of values for which the original inequality is valid.

Critical points are values of \(x\) where the polynomial either is zero or changes its sign from positive to negative, or vice versa. Finding these points is crucial in understanding the behavior of inequalities.
For example, with the inequality \((x-1)(x+1) \geq 0\), the critical points are found by setting each factor to zero. Here, solving \(x-1 = 0\) yields \(x = 1\), and \(x+1 = 0\) yields \(x = -1\).
These points allow us to divide the real number line into separate intervals. We then test these intervals to determine where the inequality is satisfied. The critical points themselves are also checked, as depending on the inequality type, they may satisfy the inequality due to being equal to zero.

A fourth root function considers the fourth root of an expression, indicated by \(\sqrt[4]{x}\). Unlike square roots, the fourth root produces positive real numbers only when its argument is non-negative, requiring expressions under the root to satisfy non-negative conditions.
In our function \(h(x) = \sqrt[4]{x^4 - 1}\), the term \(x^4 - 1\) must be zero or positive for \(h(x)\) to be defined. Therefore, this sets the requirement for inequality solving, as handled in our problem.
Understanding the characteristics of the fourth root function is essential because it dictates the conditions under which the function can be appropriately evaluated. This, in turn, impacts the overall solution, as non-negative constraints directly influence the domain of the function.

Factoring polynomials simplifies expressions by rewriting them as a product of their factors, making it easier to solve equations or inequalities. For the expression \(x^4 - 1\), recognizing difference of squares is key. This particular expression is factorable as \((x^2 - 1)(x^2 + 1)\), and further into \((x-1)(x+1)(x^2+1)\).
Factoring reveals critical point candidates from terms that could equate to zero. It's a vital step in analyzing polynomial inequalities as it divides the expression into simpler sub-units, each corresponding to potential solutions.
Applying this in our original exercise helped convert a complex rational expression into an accessible series of intervals to evaluate, leading to a thorough understanding of where the original inequality holds.