Question

Helр with math please!!))

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\sf{\dfrac{x^2-5x-36}{\:x-2}\le 0}$

♣ ᴀɴꜱᴡᴇʀ :

$\sf{\dfrac{x^2-5x-36}{x-2}\le \:0\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x\le \:-4\quad \mathrm{or}\quad \:2<x\le \:9\:\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:-4]\cup (2,\:9]\end{bmatrix}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\underline{\underline{\sf{\text { Factor } \dfrac{x^{2}-5 x-36}{x-2}}}}$

$\text { Break the expression into groups }$

$=\left(x^2+4x\right)+\left(-9x-36\right)$

$\sf{Factor\:\: out\:\:x \:\:from x^{2}+4 x: x(x+4)} \\\\Factor out -9 from -9 x-36:-9(x+4)$

$=x\left(x+4\right)-9\left(x+4\right)$

$\text { Factor out common term } x+4$

$=\left(x+4\right)\left(x-9\right)$

$\boxed{=\dfrac{\left(x+4\right)\left(x-9\right)}{x-2}}$

____________________________

$\underline{\underline{\text { Find the signs of the factors of }\: \dfrac{(x+4)(x-9)}{x-2}}}$

$\bullet\:{\text { Find the signs of } x+4$

$\begin{array}{l}x+4=0: \quad x=-4 \\x+4<0: \quad x<-4 \\x+4>0: \quad x>-4\end{array}$

$\bullet\:{\text { Find the signs of } x-9$

$\begin{array}{ll}x-9=0: & x=9 \\x-9<0: & x<9 \\x-9>0: & x>9\end{array}$

$\bullet\:{\text { Find the signs of } x-2$

$\begin{array}{ll}x-2=0: & x=2 \\x-2<0: & x<2 \\x-2>0: & x>2\end{array}$

$\text { Find the zeros of the denominator } x-2:$

$\mathrm{Add\:}2\mathrm{\:to\:both\:sides}$

$x-2+2=0+2$

$x=2$

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$\bf{Summarize\:\:in\:\:a\:\:table:}$

$\sf{\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} &x<-4 &x=-4 &-4<x<2 &x=2 & 2<x<9&x=9 &x>9\\ \cline{1-8} x+4&- &0&+ & +& +&+ &+\\ \cline{1-8}x-9 &- &- &- &- &- & 0&+\\ \cline{1-8} x-2&- &- & -&0 &+ & +&+\\ \cline{1-8}\frac{\left(x+4\right)\left(x-9\right)}{x-2} &- &0 &+&Undf& -& 0&+\\ \cline{1-8}\end{array}}$

$\mathrm{Identify\:the\:intervals\:that\:satisfy\:the\:required\:condition:}\:\le \:\:0$

$x<-4\quad \mathrm{or}\quad \:x=-4\quad \mathrm{or}\quad \:2<x<9\quad \mathrm{or}\quad \:x=9$

____________________________

$\underline{\underline{\text { Merge Overlapping Intervals }}}$

$\begin{aligned}&\text { The union of two intervals is the set of numbers which are in either interval }\\&\begin{array}{ll}x<-4 \text { or } x=-4\end{array}\end{aligned}$

$x\le \:-4$

______________

$\begin{aligned}&\text { The union of two intervals is the set of numbers which are in either interval }\\&\begin{array}{ll}x \leq-4 \text { or } 2<x<9\end{array}\end{aligned}$

$x\le \:-4\quad \mathrm{or}\quad \:2<x<9$

______________

$\begin{aligned}&\text { The union of two intervals is the set of numbers which are in either interval }\\&x \leq-4 \text { or } 2<x<9 \text { or } x=9\end{aligned}$

$x\le \:-4\quad \mathrm{or}\quad \:2<x\le \:9$

______________

$\huge\boxed{\:\:\sf{x\le \:-4\quad \mathrm{or}\quad \:2<x\le \:9}}$