Step-by-Step Solution

Step-by-Step Solution

Step 1: Use the Disjunction Property of Absolute Values

For any positive constant $c > 0$, the inequality $|A| > c$ splits into two disjoint possibilities:

Substituting $A = x^2 - 3x + 2$ and $c = 4$:

Step 2: Solve Branch 1 ($> 4$)

Find the roots of the corresponding equation $x^2 - 3x - 2 = 0$:

Numerically: $\frac{3 - 4.123}{2} \approx -0.562$ and $\frac{3 + 4.123}{2} \approx 3.562$.

Because the leading coefficient is positive ($a = 1 > 0$), the upward parabola is strictly positive outside the roots:

Step 3: Solve Branch 2 ($< -4$)

Compute the discriminant of $x^2 - 3x + 6$:

Since $\Delta < 0$ and $a = 1 > 0$, the parabola $y = x^2 - 3x + 6$ lies entirely above the $x$-axis and is strictly positive for all $x \in \mathbb{R}$. It can never be negative. Thus, Branch 2 produces no solutions ($\emptyset$).

Step 4: Union of Branches

The complete solution set is given entirely by Branch 1: