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

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