Solve the Quadratic Inequality: x²-6x+8 Greater Than Zero

Let's solve the inequality $x^2 - 6x + 8 > 0$ by breaking it down into steps:

**Step 1: Find the Roots of the Equation**
To solve the inequality, we first need to find the solutions to the equation $x^2 - 6x + 8 = 0$. This can be done by factoring or using the quadratic formula.

Let's factor the quadratic expression:
$x^2 - 6x + 8 = (x - 2)(x - 4)$

The roots of the equation are $x = 2$ and $x = 4$.

**Step 2: Test Intervals**
The roots divide the number line into three intervals: $(-\infty, 2)$, $(2, 4)$, and $(4, \infty)$. We need to test each of these intervals to see where the inequality holds.

• For $x \in (-\infty, 2)$, choose $x = 0$:
  $(0 - 2)(0 - 4) = 8$, which is positive, so $x^2 - 6x + 8 > 0$.
• For $x \in (2, 4)$, choose $x = 3$:
  $(3 - 2)(3 - 4) = -1$, which is negative, so $x^2 - 6x + 8 < 0$.
• For $x \in (4, \infty)$, choose $x = 5$:
  $(5 - 2)(5 - 4) = 3$, which is positive, so $x^2 - 6x + 8 > 0$.

**Conclusion**:
The inequality is satisfied for $x < 2$ and $x > 4$.

Thus, the solution to the inequality $x^2 - 6x + 8 > 0$ is $x < 2$ or $x > 4$.

The correct choice is $x < 2,4 < x$.