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 jatt-rockz November 5th, 2011 07:31 PM

Inequality

Solve each inequality graphically. State the solution and graph the solution:
a) (X^3) – (6x^2) +5x+12 >0
b) X/(x^2 -2x-8) >0

 Zahar November 5th, 2011 08:08 PM

Re: Inequality

1 Attachment(s)
I do not know right answer but in my head))) exist this:
X/(x^2 -2x-8) >0
for the (x^2 -2x-8) :
x = [ 2 +/- square_root( 2^2 - 4*(-8) ) ] / 2 = 1 +/- square_root( 4 + 32 )/2 =
= 1 +/- 6/2 = 1 +/- 3 = { 4, -2 } (x1 = 4 and x2 = -2)

then we can to write
x/(x^2 -2x-8) = x / [ (x - x1)*(x - x2) ] = x / [ (x - 4)*(x + 2) ]

I think this
x do not equal 4 and -2 because we will have division by zero (but we used this numbers to transform equation)

we have x from ( - infinity) to (+ infinity)
then

case ( x < - 2 ) : (-) /[ (-)*(-) ] < 0 /* because x < 0, x - 4 < 0 , x + 2 < 0 as conclusion from that :: x < - 2 */

case ( -2 < x < 0 ) : (-) /[ (-)*(+) ] > 0 /* because x < 0, x - 4 < 0 , x + 2 > 0 as conclusion from that :: -2 < x < 0 */

case ( 0 < x < 4 ) : (+) /[ (-)*(+) ] < 0 /* because x > 0, x - 4 < 0 , x + 2 > 0 as conclusion from that :: 0 < x < 4 */

case ( 4 < x ) : (+) /[ (+)*(+) ] > 0 /* because x > 0, x - 4 > 0 , x + 2 > 0 as conclusion from that :: 4 < x */

But ! You need to write it in proper form, which is demanded by teacher, book, because I do not know the rules of ...

 soroban November 5th, 2011 08:21 PM

Re: Inequality

Hello, jatt-rockz!

Quote:
 Solve each inequality graphically. State the solution and graph the solution: [color=beige]. . [/color]$a)\;x^3\,-\,6x^2\,+\,5x\,+\,12\:>\:0$

$\text{W\!e have a cubic function: }\:y \:=\:x^3\,-\,6x^2\,+\,5x\,+\,12$
[color=beige]. . [/color]When is it positive?
[color=beige]. . [/color]When is its graph above the x-axis?

The function factors:[color=beige] .[/color]$f(x) \:=\:(x\,+\,1)(x\,-\,3)(x\,-\,4)$
[color=beige]. . [/color]Its x-intercepts are:[color=beige] .[/color]$-1,\:3,\:4$

The graph looks like this:

Code:

            |             |                  *             |             | *                *       -1 * |    *          *   - - - o - + - - - o - - - o - - -       *    |      3  *  4             |       *    |             |

$\text{Solution: }\:(-1\:<\:x\:<\:3)\:\cup\:(x\:>\:4)$

$\begin{array}{cccccccccc}\text{Graph: }=&--=&o=&o=&--=&o=&-1=&3=&4 \end{array}=$

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