Question

log 0.5 (x^2-5x+6)>-1

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{log\,_{0.5}(x^2-5x+6)\;>\;-1}$

$\underline{\textbf{To find:}}$

$\textsf{Solution set of the given inequality}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{log\,_{0.5}(x^2-5x+6)\;>\;-1}$

$\textsf{This can be written as}$

$\mathsf{0.5^{log\,_{0.5}(x^2-5x+6)}\;>\;0.5^{-1}}$

$\mathsf{Using,}\;\boxed{\mathsf{a^{log\,_aN}=N}}$

$\mathsf{x^2-5x+6\;>\;\dfrac{1}{0.5}}$

$\mathsf{x^2-5x+6\;>\;2}$

$\mathsf{x^2-5x+4\;>\;0}$

$\mathsf{x^2-4x-x+4\;>\;0}$

$\mathsf{x(x-4)-1(x-4)\;>\;0}$

$\mathsf{(x-1)(x-4)\;>\;0}$

$\implies\mathsf{x\,\in\,(-\infty,1)\;\;or\;\;(4,\infty)}$

$\therefore\mathsf{Solution\;set\;is\;(-\infty,1){\cup}(4,\infty)}$

Answer 2

Answer:

• X€ [1,4]
• Step-by-step explanation:
• Step-by-step explanation:X>0
• Step-by-step explanation:X>0base <1
• Step-by-step explanation:X>0base <1X2 -5x+6<(0.5)-1
• Step-by-step explanation:X>0base <1X2 -5x+6<(0.5)-10.5^1
• Step-by-step explanation:X>0base <1X2 -5x+6<(0.5)-10.5^1X2 -5x+6<2
• Step-by-step explanation:X>0base <1X2 -5x+6<(0.5)-10.5^1X2 -5x+6<26-2=4
• Step-by-step explanation:X>0base <1X2 -5x+6<(0.5)-10.5^1X2 -5x+6<26-2=44X2 -5+4<0
• Step-by-step explanation:X>0base <1X2 -5x+6<(0.5)-10.5^1X2 -5x+6<26-2=44X2 -5+4<0( X-1 ) ( X -4)
• Step-by-step explanation:X>0base <1X2 -5x+6<(0.5)-10.5^1X2 -5x+6<26-2=44X2 -5+4<0( X-1 ) ( X -4) X€ [ 1,4 ]