Solve the inequality (2-3x/x-2)+4 > 0show complete solution to explain the answer (Show domain and range)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to solve the inequality `(2-3x)/(x-2) + 4 gt 0` , hence, you should bring the terms to a common denominator such that:

`(2-3x + 4(x-2))/(x-2) gt 0`

`(2 - 3x + 4x - 8)/(x-2) gt 0`

`(x - 6)/(x - 2) gt 0`

The frction is positive if both numerator and denominator have the same sign such that:

`x - 6gt 0 =gt x gt 6`


`x - 2 gt 0 =gt x gt 2`

The  values of x that satisfy both inequalities is `(6,oo).`

`x - 6lt 0 =gt xlt 6`


`x - 2lt 0 =gt xlt 2`

The values of x that satisfy both inequalities is `(-oo,2).`

Hence, the values of x that satisfy the given inequality are in the following intervals `(-oo,2)`  or `(6,oo).`

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