25x2>4

To solve the inequality \(25x^2 > 4\), follow these steps:

1. Rearrange the inequality: \[ 25x^2 - 4 > 0 \]

2. Factor the quadratic expression: \[ (5x - 2)(5x + 2) > 0 \]

3. Find the critical points: Set each factor equal to zero: \[ 5x - 2 = 0 \implies x = \frac{2}{5} \] \[ 5x + 2 = 0 \implies x = -\frac{2}{5} \]

4. Determine the sign of the product in each interval: The critical points divide the number line into three intervals: \((- \infty, -\frac{2}{5})\), \((- \frac{2}{5}, \frac{2}{5})\), and \((\frac{2}{5}, \infty)\).

   • Interval 1: Choose a test point \(x = -1\): \[ (5(-1) - 2)(5(-1) + 2) = (-5 - 2)(-5 + 2) = (-7)(-3) > 0 \]
   • Interval 2: Choose a test point \(x = 0\): \[ (5(0) - 2)(5(0) + 2) = (-2)(2) < 0 \]
   • Interval 3: Choose a test point \(x = 1\): \[ (5(1) - 2)(5(1) + 2) = (5 - 2)(5 + 2) = (3)(7) > 0 \]

5. Combine the results: The product \((5x - 2)(5x + 2) > 0\) is true for the intervals \((- \infty, -\frac{2}{5})\) and \((\frac{2}{5}, \infty)\).

6. State the solution: The solution to the inequality \(25x^2 > 4\) is: \[ x \in (-\infty, -\frac{2}{5}) \cup (\frac{2}{5}, \infty) \]