\(x \in (3, \infty)\) の場合、\(x = 4\) をとぶ: \(3(4)^2 - 12(4) + 9 = 9 > 0\)。

["Understanding the Inequality: Proving ( x = 4 \in (3, \infty) ) Satisfies ( 3x^2 - 12x + 9 > 0 )", "When analyzing quadratic functions and their behavior over intervals, one important skill is evaluating whether a given value satisfies a specific inequality. A classic example is determining whether ( x = 4 ) lies in the solution set of the inequality ( 3x^2 - 12x + 9 > 0 ), particularly when considering the interval ( x \in (3, \infty) ).", "### The Interval ( x \in (3, \infty) )", "The interval ( (3, \infty) ) includes all real numbers greater than 3 but does not include 3 itself. Since ( x = 4 ) is clearly greater than 3, it belongs to this domain. Plugging ( x = 4 ) into our quadratic expression:\n[\n3(4)^2 - 12(4) + 9 = 3(16) - 48 + 9 = 48 - 48 + 9 = 9\n]\nThe result is ( 9 ), which is clearly greater than 0. This confirms that ( x = 4 ) satisfies the inequality ( 3x^2 - 12x + 9 > 0 ).", "### Why ( x = 4 ) is Significant in the Interval", "The expression ( 3x^2 - 12x + 9 ) is a quadratic function opening upwards (since the coefficient of ( x^2 ) is positive). It reaches its minimum value at its vertex, found by ( x = -\frac{b}{2a} ). Here, ( a = 3 ), ( b = -12 ), so:\n[\nx = \frac{12}{2 \cdot 3} = 2\n]\nThe parabola’s minimum value is negative (evaluating at ( x = 2 )), but beyond certain points, the output increases. Since 4 lies well to the right of the vertex (and well within ( (3, \infty) )), the function grows positively. For ( x > 3 ), all outputs of ( 3x^2 - 12x + 9 ) are positive due to the upward opening and position of the function.", "### Checking the Inequality Over the Interval", "To generalize, completing the square helps fully reveal the inequality’s truth:\n[\n3x^2 - 12x + 9 = 3(x^2 - 4x) + 9 = 3\left((x - 2)^2 - 4\right) + 9 = 3(x - 2)^2 - 12 + 9 = 3(x - 2)^2 - 3\n]\nSo:\n[\n3x^2 - 12x + 9 = 3(x - 2)^2 - 3\n]\nThis confirms the function is positive when ( (x - 2)^2 > 1 ), or ( |x - 2| > 1 ), i.e., ( x < 1 ) or ( x > 3 ). Since ( x = 4 ) satisfies ( x > 3 ), it falls within the solution set.", "### Summary", "Taking ( x = 4 ) as a representative point in ( (3, \infty) ), we verified that:\n[\n3(4)^2 - 12(4) + 9 = 9 > 0\n]\nThis confirms ( x = 4 ) satisfies the inequality and illustrates how values within the interval ( (3, \infty) ) behave under this quadratic expression. Understanding these relationships supports solving similar inequalities and interpreting function behavior on bounded domains.", "---", "Keywords:\n( x \in (3, \infty) ), ( 3x^2 - 12x + 9 > 0 ), ( x = 4 ), quadratic inequality, function behavior, parabola, vertex, domain analysis, inequality verification."]