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)