For exercise, solve the equations and inequalities. Write the solutions sets to

Step 1: Analysis
Given:
${\displaystyle \frac{1}{x^2-14x+40}\le 0}$
To determine the solution sets of the given inequality.
Step 2: Simplification
Factorising the term ${\displaystyle x^2-14x+40}$.
Let ${\displaystyle p\left(x\right)=x^2-14x+40}$
${\displaystyle p\left(x\right)=x^2-14x+40}$
${\displaystyle =x^2-4x-10x+40}$
${\displaystyle =x(x-4)-10(x-4)}$
${\displaystyle =(x-10)(x-4)}$
when ${\displaystyle x=10,4.p\left(x\right)=0}$
when ${\displaystyle x\in (-\mathrm{\infty },4).p\left(x\right)>0}$
when ${\displaystyle x\in (4,10).p\left(x\right)<0}$
when ${\displaystyle x\in (10,\mathrm{\infty }).p\left(x\right)>0}$
Step 3: Solution
For ${\displaystyle \frac{1}{x^2-14x+40}\le 0}$.
${\displaystyle x^2-14x+40}$ should be less than 0.
${\displaystyle x^2-14x+40<0}$ when ${\displaystyle x\in (4,10)}$
Therefore, the solution set of the given inequality is (4,10)