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An inequality is a mathematical statement indicating that two expressions are not equal, often related by the symbols ">", "<", "≥", or "≤". In this particular example, the inequality is given by \((x^2 - 9)(x^2 - 4) > 0\). This represents regions where the product of \((x^2 - 9)\) and \((x^2 - 4)\) is greater than zero.
When solving inequalities, the aim is to find the range of values that satisfy the inequality. This involves understanding when the inequality holds true, which is either when both components are positive or both are negative, resulting in their product being positive as per the inequality condition. It is important to note the strict inequality \(>0\) here, meaning that we are specifically looking for regions where the expression is strictly above zero, not equal to zero.

Critical points are specific values that cause each factor in the inequality to change sign. To find the critical points, set each factor to zero and solve for the variable.
For the inequality \((x^2 - 9)(x^2 - 4) > 0\), the critical points come from solving \(x^2 - 9 = 0\) and \(x^2 - 4 = 0\). Solving these gives \(x^2 = 9\), resulting in \(x = \pm 3\), and \(x^2 = 4\), resulting in \(x = \pm 2\). These values divide the number line into distinct intervals, each potentially having different signs for the expression. Critical points do not satisfy the inequality themselves if the inequality is strict, as they make the entire expression zero.

Interval testing is a method to determine the sign of the inequality expression on different intervals defined by the critical points.
Divide the real number line using critical points. Here, the critical points \(-3, -2, 2,\) and \(3\) divide it into:

• \((-\infty, -3)\)
• \((-3, -2)\)
• \((-2, 2)\)
• \((2, 3)\)
• \((3, \infty)\)

Choose a test point in each interval and substitute it into the inequality:

• In \((-\infty, -3)\), \(x = -4\) gives a positive value
• In \((-3, -2)\), \(x = -2.5\) gives a negative value
• In \((-2, 2)\), \(x = 0\) gives a positive value
• In \((2, 3)\), \(x = 2.5\) gives a negative value
• In \((3, \infty)\), \(x = 4\) gives a positive value

Only intervals with positive test results satisfy the inequality \((x^2 - 9)(x^2 - 4) > 0\).

The solution set of an inequality is a representation of all the values that satisfy the given inequality. After conducting interval testing, we identify which intervals make the expression positive, as indicated by our inequality \((x^2 - 9)(x^2 - 4) > 0\).
For this particular inequality, the solution set consists of the intervals \((-\infty, -3)\), \((-2, 2)\), and \((3, \infty)\). These intervals are written together using union symbols \(\cup\) in interval notation, which is a concise format to express a range of numbers.
Therefore, the solution set in interval notation is \((-\infty, -3) \cup (-2, 2) \cup (3, \infty)\). This accurately captures the values of \(x\) for which the original product of expressions remains positive.