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Solve the inequality: -x^2+5x-6>=0

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lauradanielle | Student, Grade 10 | eNotes Newbie

Posted February 6, 2012 at 2:02 PM via web

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Solve the inequality: -x^2+5x-6>=0

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted February 6, 2012 at 3:03 PM (Answer #1)

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I suggest you to multiply the inequality by -1 to get a positive leading coefficient. You need to remember the fact that multiplication of an inequality by a negative amount reverses the sense of inequality, hence you need to solve the inequality `x^2-5x+6=lt0` .

You need to find the zeroes of quadratic such that:

`x^2-5x+6=0`

Using quadratic formula yields:

 `x_(1,2) = (5+-sqrt(25 - 24))/2 =gt x_1 = (5+1)/2 =gt x_1 = 3`

`` `x_2 = 2`

You need to calculate`x^2-5x+6`  for a value between 2 and 3 such that:

`(2.5)^2 - 5*(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25`

Notice that you get a negative value for `x^2-5x+6`  if  the values of x are in interval [2,3]

Notice that the parabola goes below x axis between 2 and 3.

Hence, the values that check the inequality are comprised by the interval [2,3].

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted February 6, 2012 at 2:32 PM (Answer #2)

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The inequality `-x^2+5x-6>=0` has to be solved

`-x^2+5x-6>=0`

=> -x^2 + 3x + 2x - 6 `>=` 0

=> -x(x - 3) + 2(x - 3) `>=` 0

=> (-x + 2)(x - 3) `>=` 0

This is possible when both the factors have the same sign

=> -x + 2 `>=` 0 and x - 3 `>=` or -x + 2 < 0 and x - 3 < 0

=> x `<=` 2 and x `>=` 3 or x > 2 and x < 3

It is not possible to have x <= 2 and x `>=` 3.

The solution of the inequality is {2 , 3}

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