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Graph Inequality
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x^{2}-5x+6=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(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\times 6}}{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, -5 for b, and 6 for c in the quadratic formula.
x=\frac{5±1}{2}
Do the calculations.
x=3 x=2
Solve the equation x=\frac{5±1}{2} when ± is plus and when ± is minus.
\left(x-3\right)\left(x-2\right)<0
Rewrite the inequality by using the obtained solutions.
x-3>0 x-2<0
For the product to be negative, x-3 and x-2 have to be of the opposite signs. Consider the case when x-3 is positive and x-2 is negative.
x\in \emptyset
This is false for any x.
x-2>0 x-3<0
Consider the case when x-2 is positive and x-3 is negative.
x\in \left(2,3\right)
The solution satisfying both inequalities is x\in \left(2,3\right).
x\in \left(2,3\right)
The final solution is the union of the obtained solutions.