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Solve the inequality |x^2-5x| < 6Show complete solution

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spock1 | Student, Undergraduate | Honors

Posted May 25, 2012 at 6:57 PM via web

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Solve the inequality |x^2-5x| < 6

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 25, 2012 at 8:06 PM (Answer #1)

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You need to remember how to solve an absolute value inequality such that:

`|x^2-5x| lt 6 =gt -6 lt x^2-5x lt 6`

You need to solve the left side inequality such that:

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

You need to find first what the zeroes of equation are such that:

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

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

`x_1 = (5+1)/2 =gt x_1 = 3`

`x_2 = 2`

Notice that the expression is positive only if `x in (-oo,2)U(3,oo).`

You need to solve the left side inequality such that:

`x^2-5x lt 6 =gt x^2- 5x -6lt 0`

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

`x_(1,2) = (5+-7)/2`

`x_1 = 6 ; x_2 = -1`

Notice that the expression is negative only if `x in (-1,6).`

You need to find the values of x that satisfy the inequality such that:

`{(-oo,2)U(3,oo)}O/ (-1,6) = (-1,2)U(3,6)`

Hence, the interval solutions to the given inequality are `(-1,2)U(3,6).`

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