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Understanding how to solve inequalities is fundamental in algebra. An inequality, unlike an equation, shows that one expression is less than, greater than, lesser or equal to, or greater or equal to another. The process begins with isolating the variable on one side of the inequality sign.

For quadratic inequalities, such as the given problem \(x^2<9\), we approach them as if they were equations to find critical values, or 'roots,' which divide the number line into intervals. It's crucial to note that these critical values may not be part of the solution but serve as endpoints for testing the intervals. Always remember to consider the inequality sign as this determines if the critical values are included \(≤, ≥\) or not included \(<, >\) in the final solution. Testing the intervals against the original inequality determines where the solution lies.

Quadratic equations are polynomial equations of the second degree, typically in the form \(ax^2 + bx + c = 0\). These equations are vital in various areas of mathematics and science. The solutions to a quadratic equation are called the 'roots' or 'zeros' and can be found through methods such as factoring, completing the square, or using the quadratic formula.

In the context of our inequality \(x^2<9\), we construct a related quadratic equation \(x^2 - 9 = 0\) to find the roots. These roots separate the number line into distinct intervals that we evaluate to determine where the original inequality holds true.

Graphing on a number line provides a visual representation of where the solutions to inequalities lie. After solving an inequality and identifying critical values, the number line is used to mark these values and test intervals.

For open inequalities \(x^2<9\), open dots or parentheses are used to show that the endpoints are not included in the solution set. Filled dots or brackets are used when the endpoints are included \(≥, ≤\). Graphing highlights the range of values that satisfy the inequality, offering a clear and immediate understanding of the solution.

Factoring is a method to solve quadratic equations that involves breaking down the equation into simpler binomials. The factored form can make it easier to identify the roots or solve inequalities.

In the step-by-step solution for \(x^2 - 9 < 0\), the quadratic was factored into \(x+3)(x-3) = 0\). This represents the product of two binomials equaling zero, which implies that at least one of the binomials must be zero. Finding the roots, \(x = -3\) and \(x = 3\), establishes the basis for testing intervals on the number line. A correct factorization is crucial for correctly determining the intervals for the solution of the inequality.