Solve the inequality 30/x-1 < x+2?

2 Answers

#x\in(\frac{-1-\sqrt{129}}{2}, 1)\cup(\frac{-1+\sqrt{129}}{2}, \infty)#

Explanation:

#\frac{30}{x-1}< x+2#

#\frac{30}{x-1}-(x+2)<0#

#\frac{30-(x+2)(x-1)}{x-1}<0#

#\frac{30-x^2-x+2}{x-1}<0#

#\frac{-x^2-x+32}{x-1}<0#

#\frac{x^2+x-32}{x-1}>0#

Using quadratic formula to find the roots of #x^2+x-32=0# as follows

#x=\frac{-1\pm\sqrt{1^2-4(1)(-32)}}{2(1)}#

#x=\frac{-1\pm\sqrt{129}}{2}#

#\therefore \frac{(x+\frac{1+\sqrt{129}}{2})(x+\frac{1-\sqrt{129}}{2})}{x-1}>0#

Solving above inequality, we get

#x\in(\frac{-1-\sqrt{129}}{2}, 1)\cup(\frac{-1+\sqrt{129}}{2}, \infty)#

Jun 26, 2018

#color(blue)((-1/2-1/2sqrt(129),1)uuu(-1/2+1/2sqrt(129),oo)#

Explanation:

#30/(x-1)< x+2#

subtract #(x+2)# from both sides:

#30/(x-1)-x-2<0#

Simplify #LHS#

#(-x^2-x+32)/(x-1)<0#

Find roots of numerator:

#-x^2-x+32=0#

By quadratic formula:

#x=(-(-1)+-sqrt((-1)^2-4(-1)(32)))/(2(-1))#

#x=(1+-sqrt(129))/-2#

#x=-1/2+1/2sqrt(129)#

#x=-1/2-1/2sqrt(129)#

For #x> -1/2+1/2sqrt(129)#

#-x^2-x+32 < 0#

For #x< -1/2+1/2sqrt(129)#

#-x^2-x+32 > 0#

For #x> -1/2-1/2sqrt(129)#

#-x^2-x+32 > 0#

For #x< -1/2-1/2sqrt(129)#

#-x^2-x+32 < 0#

Root of #x-1#

#x-1=0=>x=1#

For: #x > 1#

#x-1>0#

For #x < 1#

#x-1 < 0#

Check for:

#+/-#, #-/+#

This gives us:

#-1/2-1/2sqrt(129)< x <1#

#-1/2+1/2sqrt(129)< x < oo#

In interval notation this is:

#(-1/2-1/2sqrt(129),1)uuu(-1/2+1/2sqrt(129),oo)#