Solve the quadratic inequality \(2x^2 - 5x + 3 > 0\).

Step 1 ：To solve this inequality, we first need to find the roots of the corresponding equation \(2x^2 - 5x + 3 = 0\).

Step 2 ：We can solve this equation either by factoring, or by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case, using the quadratic formula, we get \(x = \frac{5 \pm \sqrt{(-5)^2 - 4*2*3}}{2*2} = \frac{5 \pm \sqrt{1}}{4}\), which gives us the roots \(x = \frac{3}{2}\) and \(x = 1\).

Step 3 ：Now, we divide the number line into three intervals based on these roots: \((-\infty, 1)\), \((1, \frac{3}{2})\), and \((\frac{3}{2}, +\infty)\). We then choose a test point from each interval and substitute it into the inequality to determine whether the interval is a solution to the inequality.

Step 4 ：For interval \((-\infty, 1)\), we choose \(x=0\). Substituting \(x=0\) into the inequality, we get \(2*0^2 - 5*0 + 3 > 0\), which simplifies to \(3 > 0\). This is true, so \((-\infty, 1)\) is a solution to the inequality.

Step 5 ：For interval \((1, \frac{3}{2})\), we choose \(x=\frac{5}{4}\). Substituting \(x=\frac{5}{4}\) into the inequality, we get \(2*\left(\frac{5}{4}\right)^2 - 5*\frac{5}{4} + 3 > 0\), which simplifies to \(-\frac{1}{4} > 0\). This is false, so \((1, \frac{3}{2})\) is not a solution to the inequality.

Step 6 ：For interval \((\frac{3}{2}, +\infty)\), we choose \(x=2\). Substituting \(x=2\) into the inequality, we get \(2*2^2 - 5*2 + 3 > 0\), which simplifies to \(1 > 0\). This is true, so \((\frac{3}{2}, +\infty)\) is a solution to the inequality.