inequality question

**msdonnelly** · January 15th 2007, 08:29 PM

How do you solve the inequality x²+x>1. I can not remember how to factor it. It doesnt make sense to me how you can have a single x and a negative 1

**msdonnelly** · January 15th 2007, 08:32 PM

x²+x>1. I can not remmeber how to solve this it doesnt make sense to me how you can have a single x and a negative 1. x(x+1)-1 ???

**CaptainBlack** · January 15th 2007, 09:13 PM

Originally Posted by msdonnelly

How do you solve the inequality x²+x>1. I can not remember how to factor it. It doesnt make sense to me how you can have a single x and a negative 1

The inequality can be rearranged to:

$x^2+x-1>0$ ,

but $y=x^2+x-1$ is a parabola opening upwards, so $y>0$ everywhere except between the roots (and at the roots of course)
of $x^2+x-1=0$ .

The roots of this quadratic are $\frac{-1 \pm \sqrt{5}}{2}$ , so the inequality is satisfied if $x<\frac{-1 - \sqrt{5}}{2}$ or $x>\frac{-1 + \sqrt{5}}{2}$

RonL

**earboth** · January 16th 2007, 01:33 AM

Originally Posted by msdonnelly

x²+x>1. I can not remmeber how to solve this it doesnt make sense to me how you can have a single x and a negative 1. x(x+1)-1 ???

Hello,

this question has been answered twice - and I don't like to work for the waste-bin.

EB

**topsquark** · January 16th 2007, 04:07 AM

Originally Posted by msdonnelly

x²+x>1. I can not remmeber how to solve this it doesnt make sense to me how you can have a single x and a negative 1. x(x+1)-1 ???

$x^2 + x > 1$

$x^2 + x - 1 > 0$

(Take a look at the graph I have attached.)

We are looking for the portion of the x-axis where the graph is above the x-axis.

We do this algebraically by the following process.

Solve $x^2 + x - 1 = 0$
$x = \frac{-1 \pm \sqrt{5}}{2}$ according to the quadratic formula.

Now break the x-axis into three parts:
$(- \infty, \frac{-1 - \sqrt{5}}{2} )$
$( \frac{-1 - \sqrt{5}}{2} , \frac{-1 + \sqrt{5}}{2} )$
$(\frac{-1 + \sqrt{5}}{2}, \infty )$

We want to test the inequality on each of these intervals. The idea is to pick the easiest number in the interval to do the test on, so for example pick -100000000000000 in the first interval, pick 0 in the second, and pick 100000000000000 in the last interval. (Or any other numbers you like.)
$(- \infty, \frac{-1 - \sqrt{5}}{2} )$ : $x^2 + x - 1 > 0$ Check!

$( \frac{-1 - \sqrt{5}}{2} , \frac{-1 + \sqrt{5}}{2} )$ : $x^2 + x - 1 < 0$ Nope!

$(\frac{-1 + \sqrt{5}}{2}, \infty )$ : $x^2 + x - 1 > 0$ Check!

So the solution set is $(- \infty, \frac{-1 - \sqrt{5}}{2} ) \cup (\frac{-1 + \sqrt{5}}{2}, \infty )$ .

-Dan

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