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x^{2}-3x+2=0
To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 1\times 2}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -3 for b, and 2 for c in the quadratic formula.
x=\frac{3±1}{2}
Do the calculations.
x=2 x=1
Solve the equation x=\frac{3±1}{2} when ± is plus and when ± is minus.
\left(x-2\right)\left(x-1\right)<0
Rewrite the inequality by using the obtained solutions.
x-2>0 x-1<0
For the product to be negative, x-2 and x-1 have to be of the opposite signs. Consider the case when x-2 is positive and x-1 is negative.
x\in \emptyset
This is false for any x.
x-1>0 x-2<0
Consider the case when x-1 is positive and x-2 is negative.
x\in \left(1,2\right)
The solution satisfying both inequalities is x\in \left(1,2\right).
x\in \left(1,2\right)
The final solution is the union of the obtained solutions.