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x^{2}-7x+12=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(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 1\times 12}}{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, -7 for b, and 12 for c in the quadratic formula.
x=\frac{7±1}{2}
Do the calculations.
x=4 x=3
Solve the equation x=\frac{7±1}{2} when ± is plus and when ± is minus.
\left(x-4\right)\left(x-3\right)\leq 0
Rewrite the inequality by using the obtained solutions.
x-4\geq 0 x-3\leq 0
For the product to be ≤0, one of the values x-4 and x-3 has to be ≥0 and the other has to be ≤0. Consider the case when x-4\geq 0 and x-3\leq 0.
x\in \emptyset
This is false for any x.
x-3\geq 0 x-4\leq 0
Consider the case when x-4\leq 0 and x-3\geq 0.
x\in \begin{bmatrix}3,4\end{bmatrix}
The solution satisfying both inequalities is x\in \left[3,4\right].
x\in \begin{bmatrix}3,4\end{bmatrix}
The final solution is the union of the obtained solutions.