3x ^ 2 - 5x + 4 < 0

To solve the quadratic inequality \(3x^2 - 5x + 4 < 0\), we first need to analyze the quadratic expression.

Step 1: Find the roots of the quadratic equation

We can find the roots of the corresponding quadratic equation \(3x^2 - 5x + 4 = 0\) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a = 3\), \(b = -5\), and \(c = 4\).

Step 2: Calculate the discriminant

First, calculate the discriminant (\(D\)): \[ D = b^2 - 4ac = (-5)^2 - 4(3)(4) = 25 - 48 = -23 \]

Step 3: Interpret the discriminant

Since the discriminant is negative (\(D < 0\)), this means that the quadratic equation \(3x^2 - 5x + 4 = 0\) has no real roots. As a result, the quadratic function does not cross the x-axis.

Step 4: Determine the sign of the quadratic expression

Next, since the leading coefficient \(a = 3\) is positive, the quadratic \(3x^2 - 5x + 4\) opens upwards. Therefore, the entire quadratic expression will be greater than zero for all \(x\).

Conclusion

The inequality \(3x^2 - 5x + 4 < 0\) has no solutions, since there are no x-values for which the quadratic expression is negative.

In summary: \[ \text{The solution to } 3x^2 - 5x + 4 < 0 \text{ is } \text{no solutions}. \]