(5х-2)(х+6)>0

To solve the inequality \( (5x - 2)(x + 6) > 0 \), we can follow these steps:

1. Find the roots of the equation \( (5x - 2)(x + 6) = 0 \): \[ 5x - 2 = 0 \quad \Rightarrow \quad 5x = 2 \quad \Rightarrow \quad x = \frac{2}{5} \] \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \]

2. Determine the critical points: The points where the expression is either zero or undefined are \( x = -6 \) and \( x = \frac{2}{5} \).

3. Set up intervals based on the critical points: The critical points divide the number line into three intervals:

   • \( (-\infty, -6) \)
   • \( (-6, \frac{2}{5}) \)
   • \( (\frac{2}{5}, \infty) \)

4. Test a point in each interval to determine where the inequality holds:

   • For the interval \( (-\infty, -6) \), choose \( x = -7 \): \[ (5(-7) - 2)(-7 + 6) = (-35 - 2)(-1) = (-37)(-1) = 37 > 0 \quad \text{(satisfies the inequality)} \]

   • For the interval \( (-6, \frac{2}{5}) \), choose \( x = 0 \): \[ (5(0) - 2)(0 + 6) = (-2)(6) = -12 < 0 \quad \text{(does not satisfy the inequality)} \]

   • For the interval \( (\frac{2}{5}, \infty) \), choose \( x = 1 \): \[ (5(1) - 2)(1 + 6) = (5 - 2)(7) = (3)(7) = 21 > 0 \quad \text{(satisfies the inequality)} \]

5. Combine the results: The intervals where the product is greater than zero are \( (-\infty, -6) \) and \( (\frac{2}{5}, \infty) \).

6. Write the solution: The solution to the inequality \( (5x - 2)(x + 6) > 0 \) is: \[ x \in (-\infty, -6) \cup \left( \frac{2}{5}, \infty \right) \]

So, the final answer is: \[ x < -6 \quad \text{or} \quad x > \frac{2}{5} \]