|x^2-6x-9|<2 solve the inequalities

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use the absolute value property such that:

`|x^2-6x-9| lt 2 =gt -2lt x^2-6x-9 lt 2`

Hence, you should solve the system of inequalities such that:

`x^2-6x-9 lt 2 =gt x^2-6x-9-2 lt 0 =gt x^2-6x-11 lt 0`

You should solve the equation `x^2-6x-11 = 0`  first such that:

`x_(1,2) = (6+-sqrt(36+44))/2`

`x_(1,2) = (6+-sqrt(80))/2 `

`x_(1,2) = (6+-4sqrt5)/2`

`x_1 = 3+2sqrt5 ; x_2 = 3-2sqrt5`

Notice that the expression is negative for `x in (3-2sqrt5 ; 3+2sqrt5)`  as the graph below proves it:

You need to solve the next inequality `-2lt x^2-6x-9`  such that:

`x^2-6x-9 +2 gt 0 `

`x^2-6x-7gt0 `

`x^2-6x-7=0`

`x_(1,2) = (6+-sqrt(36+28))/2 `

`x_(1,2) = (6+-sqrt64)/2`

`x_(1,2) = (6+-8)/2 =gt x_1 = 7 ; x_2 = -1 `

Notice that the expression is positive for `x in (-oo,-1) U (7, oo)`  as the graph below proves it:

You need to select the common solution such that:

`(3-2sqrt5 ; 3+2sqrt5) nn {(-oo,-1) U (7, oo)}`

Hence, according to the graphs above, the common solution lies between the red and green curves, hence, the values of x in intervals `(3-2sqrt5 ; -1)U(7;3+2sqrt5)`  verify the inequality.

Sources:

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