x^2-2x-8>0

I know that the answer is x>4 or x<-2. My question is why the sign is reversed if there is no multiplication or division on either side?

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- Aug 18th 2005, 10:04 AMkdstalnakerquadratic inequality
x^2-2x-8>0

I know that the answer is x>4 or x<-2. My question is why the sign is reversed if there is no multiplication or division on either side? - Aug 18th 2005, 11:20 AMrgep
Consider x^2-2x-8 = (x-4)(x+2). This is > 0 if either x-4 and x+2 are both > 0 or both < 0. The first case leads to x>4 and x>-2, which is just x>4; the second to x<4 and x<-2, which is x<-2. The reversal of sign comes from this second case where you consider the possibility that two negative factors multiply to give a positive value.

- Aug 19th 2005, 12:16 AMticbol
Here is one way.

Get the critical points of the inequality, or the points when the graph of the inequality crosses the x-axis. You treat the inequality as an equation only---meaning, forget about the < and > signs. Consider only the = sign.

Then, these critical points subdivides the x-axis into intervals of the x-values.

Then, test/check the inequality in these intervals. The solution/solutions of the inequality are the intervals where the inequality is valid/true.

x^2 -2x -8 > 0

x^2 -2x -8 = 0

(x-4)(x+2) = 0

x = 4, or -2

So there are 3 inervals:

(-infinity,-2), (-2,4), and (4,infinity)

Test the x^2 -2x -8 > 0 in the (-infinity,-2) interval.

Say, x = -3.

(-3)^2 -2(-3) -8 >? 0

9 +6 -8 >? 0

7 >? 0

Yes, so, this interval is a solution. -----***

Test the x^2 -2x -8 > 0 in the (-2,4) interval.

Say, x = 0.

(0)^2 -2(0) -8 >? 0

0 -0 -8 >? 0

-8 >? 0

No, so, this interval is not a solution.

Test the x^2 -2x -8 > 0 in the (4,infinity) interval.

Say, x = 7.

(7)^2 -2(7) -8 >? 0

49 -14 -8 >? 0

27 >? 0

Yes, so, this interval is a solution also. -----***

Therefore,

x = (-infinity,-2) ---or x < -2

x = (4, infinity) ----or x > 4

---------------

What about the critical x's themselves, when x = -2, and when x=4?

The inequality does not say "greater than or equal to" zero, hence x cannot equal -2 0r 4.

If the inequality were x^2 -2x -8 >= 0,

then x <= -2 or x >= 4.