- Identify the roots of the quadratic expression \(5(x-2)(x+4) = 0\) as \(x = -4\) and \(x = 2\).
- Recognize that the quadratic expression represents an upward-opening parabola since the coefficient of \(x^2\) is positive.
- Conclude that the expression is positive outside the roots, meaning \(x < -4\) or \(x > 2\).
- The solution set is \(\boxed{\{x \mid x < -4 \text{ or } x > 2\}}\).
Explanation
- Analyzing the Inequality
The given inequality is \(5(x-2)(x+4) > 0\). We are dealing with a quadratic expression of the form \(f(x) = 5(x-2)(x+4)\). Since the coefficient of the \(x^2\) term (which is 5) is positive, the graph of this function is a parabola that opens upwards. The roots of the expression are the values of \(x\) that make \(f(x) = 0\). Setting the factors equal to zero, we find the roots are \(x = 2\) and \(x = -4\).
- Determining the Intervals of Positivity
Because the parabola opens upwards, the values of the function are negative between its roots (where the parabola dips below the x-axis) and positive outside of its roots (where the parabola rises above the x-axis). We are looking for the solution set where the expression is strictly greater than zero, i.e., \(f(x) > 0\). This occurs when \(x\) is less than the smaller root or greater than the larger root.
- Concluding the Solution Set
The roots we identified are \(-4\) and \(2\). Therefore, the expression \(5(x-2)(x+4)\) is greater than zero when \(x < -4\) or when \(x > 2\). Comparing this result with the given choices, we see it matches the second option: \(\{x \mid x < -4 \text{ or } x > 2\}\).
Examples
In physics, if you are modeling the trajectory of a projectile with a height equation, you might need to determine when the object is above a certain level or when it hits the ground. Solving quadratic inequalities helps identify the time intervals where the height is positive, which is crucial for determining how long an object remains in the air or when it crosses a specific barrier. This mathematical technique is a foundation for understanding motion and boundary conditions in various scientific fields.