 # For exercise, solve the equations and inequalities. Write the solutions sets to Anonym 2021-10-26 Answered
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)