 # If ${\log _{\dfrac{1}{2}}}\left( {{x^2} - 5x + 7} \right) > 0$, then find the exhaustive range of values of x.A. $\left( { - \infty ,{\rm{ }}2} \right) \cup \left( {3,{\rm{ }}\infty } \right)$B. $\left( {2,3} \right)$C. $\left( { - \infty ,{\rm{ }}1} \right) \cup \left( {1,{\rm{ }}2} \right) \cup \left( {2,{\rm{ }}\infty } \right)$D. None of these

Last updated date: 26th Jan 2023
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Hint: In order to solve the question, first check whether the given term is positive. Next, write the condition for the given term to be positive. Finally, factorize and find the exhaustive range.

Formula used:
If $\left( {x - a} \right)\left( {x - b} \right) < 0$, then $x \in \left( {a,b} \right)$

Complete step-by-step solution :
Given that
${\log _{\dfrac{1}{2}}}\left( {{x^2} - 5x + 7} \right) > 0$
If the base of the logarithm is less than 1 and the value of the logarithm is greater than 0, then the function must be less than 1.
That is the given value is positive. For the given value to be positive
$\left( {{x^2} - 5x + 7} \right) < 1$
${x^2} - 5x + 6 < 0$
$\left( {x - 2} \right)\left( {x - 3} \right) < 0$
That is
$x \in \left( {2,3} \right)$
Hence option B is the correct answer.