Answer:
the first one [tex]\:x\le \:-3\quad \mathrm{or}\quad \:2\le \:x<7[/tex]
Step-by-step explanation:
Factor [tex]x^2+x-6[/tex] :
Break the expression into groups
[tex]\left(x^2-2x\right)+\left(3x-6\right)[/tex]
[tex]\mathrm{Factor\:out\:}x\mathrm{\:from\:}x^2-2x\mathrm{:\quad}[/tex]
[tex]x\left(x-2\right)[/tex]
[tex]\mathrm{Factor\:out\:}3\mathrm{\:from\:}3x-6\mathrm{:\quad}[/tex]
[tex]3\left(x-2\right)[/tex]
[tex]x\left(x-2\right)+3\left(x-2\right)[/tex]
[tex]\mathrm{Factor\:out\:common\:term\:}x-2[/tex]
[tex]\left(x-2\right)\left(x+3\right)[/tex]
[tex]\frac{\left(x-2\right)\left(x+3\right)}{x-7}\\[/tex]
[tex]\frac{\left(x-2\right)\left(x+3\right)}{x-7}\le \:0[/tex]
[tex]\mathrm{Find\:the\:signs\:of\:the\:factors\:of\:}\frac{\left(x-2\right)\left(x+3\right)}{x-7}[/tex]
[tex]\mathrm{Identify\:the\:intervals\:that\:satisfy\:the\:required\:condition:}\:\le \:\:0[/tex]
[tex]x<-3\quad \mathrm{or}\quad \:x=-3\quad \mathrm{or}\quad \:x=2\quad \mathrm{or}\quad \:2<x<7[/tex]
Merge Overlapping Intervals
[tex]x\le \:-3\quad \mathrm{or}\quad \:2\le \:x<7[/tex]
